Can the president ban travel here by foreign nationals?

Can President Trump do this?

Short answer: Yes.
Long answer: Heck yes.

I've covered this before, but under both the Constitution and federal law, there is no restriction on what foreign nationals the nation's president may prohibit entry into the United States. If the president wants to bar entry by left-handed redheads born on odd-number-dated Thursdays, s/he may do so with complete impunity.

Since the weekend, when President Trump banned entry into the United States from seven named nations (the same seven, btw, that President Obama also restricted), many commentators have claimed that religious discrimination in immigration is prohibited by the First Amendment to the Constitution, either the establishment clause or the free-practice clause, or both. They claim that Trump has ordered a "Muslim ban" since all seven of the countries are nationally Islamic.

Well, it's a pretty curious Muslim ban that leaves Indonesia unaffected, which has 202 million Muslims, the largest Muslim population in the world (IIRC, more than all Muslims in all Arab countries). Or India, for that matter, which has 172 million Muslims.

Here are the seven nations named:

1. Syria, pop. 22.85 million
2. Iran, 77.5 million
3. Iraq, 33.4 million
4. Libya, 6.2 million
5. Somalia, 10.5 million
6. Sudan, 38 million
7. Yemen, 24.4 million

Number of people affected by the temporary ban: 212.8 million, or not many more than the population of Indonesia alone. So if this order is a "Muslim ban," it's failing miserably.

Set aside for now the question of wisdom or the humanity of the order. The question in this post simply is whether it would pass legal and Constitutional muster.

In fact, it already meets both. And there are ample precedents in both law and court cases that say so.

Here is the key and central point: Foreign Nationals outside the US have no Constitutional protections

Constitutional protections apply to the persons who are US citizens or under the jurisdiction of the United States. Jurisdiction is physical space, geographically defined. It is the territories of the 50 states of the union plus territories of non-states such as Guam or Puerto Rico, governed by the US federal government. (There are complexities regarding the Constitutional protections of US citizens outside US jurisdiction, but that's not what Trump is talking about.)

Persons who are not US citizens and who are not physically present on US territory are called "foreigners." There are about seven billion of them. They have no Constitutional protections at all for the simple reason that they are under the sovereign authority of the entity wherever they may be located.

In the 1972 case, Kleindienst v. Mandel, The US Supreme Court ruled that Belgian national Ernest Mandel, who had applied for entry into the US, had no Constitutional right to enter. The ruling said in part,

It is clear that Mandel personally, as an unadmitted and nonresident alien, had no constitutional right of entry to this country as a nonimmigrant or otherwise.

That means that the US federal government may, for any reason it chooses, bar entry onto US territory by anyone it chooses, for any reason it chooses, whether one person or many.

But wait, there's more! The president enjoys that actual power right now without consulting with Congress in the slightest. Title 8, Section 1182 of the U.S. Code provides in relevant part:

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.

But wait, there's still more! In Kerry v. Din, 2015, the US Supreme Court,

... clearly separated the rights for people inside and outside the U.S.

“Due process applies to people who are in the United States, whether you are a citizen, not a citizen and you cross the border without inspection,” [immigration attorney] Bretz said. “It does not apply to people abroad."

USA Today has a pretty good and nonpartisan summary discussion of Trump's order. I would only pick the nit that in the article's "Is it legal?" section, it seems to disregard altogether that Trump's order has a duration of only three months. That's an essential aspect of the legal issue because, as the article points out, Congress, not the president, sets enduring policy on immigration. However, the president sets policy on visa issuance apart from immigration.