Profiling and/or banning a population based on religion, ethnicity, are illegal under the constitution. The United States constitution guarantees every person equal protection under the law. President Trump cannot ban Muslims from coming to the United States. This ban will be challenged in courts and overturned by the Supreme Court, if it reaches it.
Banning a particular race or religion goes against what America was founded on. Freedom of religion, not just one religion; equality for all races, not just one race. Donald Trump is treading on fragile grounds and will ultimately face consequences for his actions that are basically unconstitutional and infringing on rights.
No, profiling immigrants on the basis of faith and nationality may not be a violation of the constitution. It's easy to hear distasteful ideas and actions and say that they are a violation of the constitution. However, the Constitution is about American citizens. It doesn't necessarily apply to immigrants. So, while the idea is definitely distasteful and against our American way, it is not necessarily unconstitutional.
In the constitution, it says that the government can't discriminate against citizens. But when it comes to non-citizens, all bets are off. Instead, the United States has been discriminating against immigrants for years. They have quotas for how many people can come from each country in a year. This is just a quota of zero.