The basic American Principle is the understanding that the individual is sovereign, individual rights are broad and should be inviolate, and that the only legitimate purpose of government is to protect the exercise of every individual's right. Unfortunately, few Americans understand this. Many fewer understand this in most countries of the world. What is more, not only do they recognize few if any individual rights in many countries, but even compared to most Americans of an authoritarian bent, they believe in much more brutal suppression of individual rights. In some countries, the majority of the people do not believe in even such simple and fundamental freedoms as freedom of speech, freedom of press, or freedom of conscience. Freedom of conscience means one is free to exercise a different religious belief or none at all. One is free to exercise a moral belief which has no basis in any religion. Our immigration policies should recognize the severity of the problems for our society that these differences in belief, which are often very firmly held, will have.
We should welcome everyone who believes in the American Principle as an immigrant. It is reasonable to allow people to come to the United States as immigrants who are good candidates to learn this principle better in time and provided they are not too firmly set in beliefs that are anathema to this American Principle. I see no reason to welcome immigrants who are adamantly opposed to the American Principle. I do not see a reason why one is obliged to welcome someone into one's home who means to abridge your individual rights. Yes, of course, my freedom of association and my property rights allow me to select who will enter my literal home. There is some difference as to the limits of restriction one can impose in one's country without being intolerant and unwelcoming to different viewpoints and ideas. As long as the differences of opinion and belief do not consist of a permission or a moral imperative to initiate the use of force to violate the rights of another individual, our society should be welcoming to newcomers who many have many very divergent viewpoints. But we do not as a People who control a government which is supposed to protect everyone's rights have a moral obligation to welcome individuals to the United States who want to suppress the rights of other individuals by the use of force. Indeed, it is irrational to do so.
Every individual right exists in the context that each individual has that right only so long as he or she does not deny the exercise of that same right to others by the use of force. I understand that many Objectivists and many libertarians believe that a complete open door immigration policy is a moral requirement. That belief fails to understand the context for individual rights which I have just named. As I have pointed out before in The Pre-Conditions for Religious Freedom Unmet by Islam, there is no right to emulate The Profit Mohammed's use of force to spread Islam and to prevent anyone from giving up the religion. There is no right to establish a government based on Islam. There is no right to restrict criticism of Islam in speech or in the press.
The following article by Nick Saffran of AEI is quite interesting in addressing the problem of welcoming immigrants from many or most of the Muslim majority nations: Terrorism Is Not The Only Reason To Be Skeptical of Muslim Immigration.