Is is Constitutional to Prevent the Immigration of Muslims or Certain Nationalities?

The short answer is,

"if you (like me) don’t want Trump to block Muslims from entering the United States, then stop him from getting elected president. Don’t depend on the Constitution, Congress, or the courts." ("But President Trump Couldn’T Exclude Muslims By Himself, Could He?", Eric Posner, 8 Dec 2015)

 It may be a good idea for a President to have the power to do this... maybe. When a President bans people solely based on their religion or their nationality, they are on thin ice.

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof... (First Amendment)

If we do not stand up for the rights of those outside of our "tribe", we are opening up ourselves to being abused by that same government.

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Americans are not possessed of more natural rights than non-Americans; rather, we enjoy more opportunities to exercise those rights because the government is theoretically restrained by the Constitution, which explicitly recognizes the natural law. That recognition is articulated in the Ninth Amendment, which declares that the enumeration of certain rights in the Constitution shall not be used by the government as an excuse to deny or disparage other unnamed and unnamable rights retained by the people. ("Immigration Is a Natural Right: Nativism is the arch-enemy of the freedom to travel", Andrew Napolitano, 31 Jan 2013)

George Washington sent a letter to the Hebrew congregation at Newport

May the children of the stock of Abraham who dwell in this land continue to merit and enjoy the good will of the other inhabitants — while every one shall sit in safety under his own vine and fig tree and there shall be none to make him afraid. 

May the father of all mercies scatter light, and not darkness, upon our paths, and make us all in our several vocations useful here, and in His own due time and way everlastingly happy. ("Letter to the Hebrew Congregation at Newport", George Washington, 21 Aug 1790)

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Those who would give up essential Liberty, to purchase a little temporary Safety, deserve neither Liberty nor Safety. ("Reply to the Governor", Benjamin Franklin, 11 Nov 1755)