As long as media and politicians are saying "Islam is the religion of peace" and burying the underlying problems, no progress seems to be made. It may indeed be that the majority of Muslims are peaceful, but it is also undeniable that:

(1) polls show a sizable minority support jihad and even suicide bombing

(2) results world wide show wherever there are Muslims some of them commit terrorist acts,

(3) world wide wherever Muslims gain large numbers, they act politically and violently in ways that harm other peoples rights.

It is sometimes said, in defense to discussion of passages in the Koran that call for violence and evil, that there are bad passages in the Bible. But Christianity has gone through a reformation. Islam hasn't. Perhaps a ban on Muslim immigration will spark a reformation in Islam.

It may indeed be that the majority of Muslims are peaceful, but

(1) polls show a sizable minority support jihad and even suicide bombing and that a sizable minority of Muslims in the US hate America.

(2)  World wide data indicate wherever there are Muslims some of them commit terrorist acts and almost all terrorist acts are committed by Muslims,

(3) world wide wherever Muslims gain large numbers, they act both politically and violently in ways that harm other peoples rights. They deny freedom of speech and religion and sexual choices to others. They take actions that a wide spectrum of non-Muslim opinion would normally view as rape and murder.

At 1% muslim population, a nation typically experiences some terrorism.

At 2% to 5%, they begin to proselytize from other ethnic minorities and disaffected groups, often with major recruiting from the jails and among street gangs.

From 5% on, they exercise an inordinate influence in proportion to their percentage of the population. At this point, they are seen to work to get the ruling government to allow them to rule themselves (within their ghettos) under Sharia, the Islamic Law. The ultimate goal of Islamists is to establish Sharia law over the entire world.

When Muslims approach 10% of the population, they tend to increase lawlessness as a means of complaint about their conditions.

After reaching 20%, nations experience hair-trigger rioting, jihad militia formations, sporadic killings, and the burnings of Christian churches and Jewish synagogues. At 40%, nations experience widespread massacres, chronic terror attacks, and ongoing militia warfare...

Update: this link has been censored: http://www.examiner.com/article/as-muslim-population-grows-what-can-happen-to-a-society ;

 but the gist of the article is still available here: https://247sports.com/college/usc/Board/59419/Contents/As-Muslim-population-grows-what-can-happen-to-a-society-41331804/

http://www.infowars.com/islamic-terrorism-why-there-is-none-in-japan/ 

What the constitution says on the subject would be in the First Amendment: Amendment I Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances. http://billofrights.org    

The law Congress has actually made is: 8 US Code 1182 section (f) which reads:
(f) Suspension of entry or imposition of restrictions by President Whenever the President finds that the entry of any aliens or of any class of aliens into the United States would be detrimental to the interests of the United States, he may by proclamation, and for such period as he shall deem necessary, suspend the entry of all aliens or any class of aliens as immigrants or nonimmigrants, or impose on the entry of aliens any restrictions he may deem to be appropriate. Whenever the Attorney General finds that a commercial airline has failed to comply with regulations of the Attorney General relating to requirements of airlines for the detection of fraudulent documents used by passengers traveling to the United States (including the training of personnel in such detection), the Attorney General may suspend the entry of some or all aliens transported to the United States by such airline. http://www.law.cornell.edu/uscode/text/8/1182    

This doesn't seem to be a law respecting an establishment of religion, thus disallowed by the First Amendment. And its use by the President to exclude Muslims wouldn't make it so. In World War 2 citizens of enemy nations were denied entry. Jimmy Carter denied Iranians entry. Also various Supreme Court decisions have established the principle that the Constitution is not a suicide pact. Muslims do pose a danger. Moderate Muslims admit that they can't tell the dangerous ones, and the government agencies charged with vetting immigrants have all said they can't either. A Presidential decision that admitting them would be detrimental to the interests of the United States would not be unreasonable. Also, we are talking here of non-citizens and residents. It isn't clear why they have any right at all under the first amendment. There are Supreme Court precedents supporting the lack of constitutional rights of foreigners as well.

The President takes an oath to preserve and protect the constitution. That's arguably his first duty. Islam is arguably an ideology that is at war with the constitution. If enough Muslims move here or they convert enough Americans, they will likely subvert the constitution. 

http://medium.com/@faisalalmutar/humanist-and-former-moderate-muslim-on-how-to-tell-a-moderate-muslim-from-a-radical-muslim-d01e9366e7b8#.ce7pxvaaq 

(1) If Muslims are prohibited from immigration, Islam may realize it needs to clean up its act, to the benefit of all.

(2) To continue to allow Muslim immigration is to acquiesce in their continuing expansion that threatens the rights of American citizens through their increased political power.

(3) To continue to allow Muslim immigration is to accept that there will be terror incidents and dead Americans as a result.

(4) There is no moral or constitutional obligation for the US to allow foreigners to settle here, nor to refrain from deciding who may settle here based on their ideology, religion, or customs.

(5) neither the Koran nor current Sharia Grant equal rights to infidels. https://t.co/2nVJlv6p2G

They don't have a moral or constitution right to be admitted, and the voters would likely be better off in the long run without them.

Many Muslim Immigrants haven't been assimilating in Europe or here.

If continue business as usual, you can count on some terror acts. You can also count on political encroachment as their numbers become larger, locally and if they become substantially larger, more nationally. If Muslim immigration and/or birth brings them into big numbers, they can be expected to pass laws and influence culture against rights Americans have enjoyed. 

 A first-of-its-kind survey of the hordes of Syrian refugees entering Europe found 13% support the Islamic State. The poll should raise alarms about the risks posed by the resettlement of 10,000 refugees in the U.S.

The poll of 900 Syrian refugees by the Arab Center for Research and Policy Studies also found that another 10% of the displaced Syrians have a lukewarm, but not entirely negative, view of the terror group. That means 23% — or almost 1 in 4 — could be susceptible to ISIS recruitment.

It also means as many 2,500 of the 10,000 Syrian refugees that the Obama administration is resettling inside American cities are potential terrorist threats.

http://www.investors.com/politics/editorials/poll-13-of-syrian-refugees-support-isis/

If Muslim population here increases dramatically, there will be calls for them to live under Sharia law

51% of U.S. Muslims want option of living uner Sharia Polling Co Poll.; 60% of young Muslims more loyal to Islam than to U.S. Pew poll.

• Muzammil Siddiqi, chairman of both the Fiqh Council of North America, which dispenses Islamic rulings, and the North American Islamic Trust, which owns most of the mosques in the U.S.: “As Muslims, we should participate in the system to safeguard our interests and try to bring gradual change, (but) we must not forget that Allah’s rules have to be established in all lands, and all our efforts should lead to that direction.”

• Omar Ahmad, co-founder of the Council on American-Islamic Relations, the top Muslim lobby group in Washington: “Islam isn’t in America to be equal to any other faith, but to become dominant. The Quran should be the highest authority in America, and Islam the only accepted religion on Earth.”

• CAIR spokesman Ibrahim Hooper: “I wouldn’t want to create the impression that I wouldn’t like the government of the United States to be Islamic sometime in the future.”

• Imam Siraj Wahhaj, director of the Muslim Alliance in North America: “In time, this so-called democracy will crumble, and there will be nothing. And the only thing that will remain will be Islam.”

• Imam Zaid Shakir, co-founder of Zaytuna College in Berkeley, Calif.: “If we put a nationwide infrastructure in place and marshaled our resources, we’d take over this country in a very short time. . . . What a great victory it will be for Islam to have this country in the fold and ranks of the Muslims.”…

https://www.jihadwatch.org/2015/10/51-of-u-s-muslims-want-sharia-60-of-young-muslims-more-loyal-to-islam-than-to-u-s 

Other poll results further supporting the above are found here:
https://pjmedia.com/blog/shocking-polls-show-what-u-s-muslims-think-of-u-s-laws/ 

See figure.

https://morningconsult.com/2016/08/18/american-immigrants-back-trump-ideological-test-poll-shows/ 
Morning consult poll reports immigrants have almost as strong support for this plan as all voters

What the constitution says on the subject would be in the First Amendment: Amendment I Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances. http://billofrights.org    

The law Congress has actually made is: 8 US Code 1182 section (f) which reads:
(f) Suspension of entry or imposition of restrictions by President Whenever the President finds that the entry of any aliens or of any class of aliens into the United States would be detrimental to the interests of the United States, he may by proclamation, and for such period as he shall deem necessary, suspend the entry of all aliens or any class of aliens as immigrants or nonimmigrants, or impose on the entry of aliens any restrictions he may deem to be appropriate. Whenever the Attorney General finds that a commercial airline has failed to comply with regulations of the Attorney General relating to requirements of airlines for the detection of fraudulent documents used by passengers traveling to the United States (including the training of personnel in such detection), the Attorney General may suspend the entry of some or all aliens transported to the United States by such airline. http://www.law.cornell.edu/uscode/text/8/1182    

This doesn't seem to be a law respecting an establishment of religion, thus disallowed by the First Amendment. And its use by the President to exclude Muslims wouldn't make it so. In World War 2 citizens of enemy nations were denied entry. Jimmy Carter denied Iranians entry. Also various Supreme Court decisions have established the principle that the Constitution is not a suicide pact. Muslims do pose a danger. Moderate Muslims admit that they can't tell the dangerous ones, and the government agencies charged with vetting immigrants have all said they can't either. A Presidential decision that admitting them would be detrimental to the interests of the United States would not be unreasonable. Also, we are talking here of non-citizens and residents. It isn't clear why they have any right at all under the first amendment. There are Supreme Court precedents supporting the lack of constitutional rights of foreigners as well.

The President takes an oath to preserve and protect the constitution. That's arguably his first duty. Islam is arguably an ideology that is at war with the constitution. If enough Muslims move here or they convert enough Americans, they will likely subvert the constitution. 

(1) If Muslims are prohibited from immigration, Islam may realize it needs to clean up its act, to the benefit of all.

(2) To continue to allow Muslim immigration is to acquiesce in their continuing expansion that threatens the rights of American citizens through their increased political power.

(3) To continue to allow Muslim immigration is to accept that there will be terror incidents and dead Americans as a result.

(4) There is no moral or constitutional obligation for the US to allow foreigners to settle here, nor to refrain from deciding who may settle here based on their ideology, religion, or customs.

http://www.infowars.com/islamic-terrorism-why-there-is-none-in-japan/

https://truthsift.com/search_view?topic=&id=529&nid1=5336&nid2=5335&fs=0