How can I assist other graduate students who may feel increasingly "unwelcome" due to their countries of origin? I work in a research group with a large number of international graduate students. Recent news in my country (the United States) has been deeply unsettling for some of these students, especially the ones who hail from "unwelcome" countries (see here).

As a U.S. citizen, I (personally) feel guilty and embarrassed over the current goings-on (nor do I agree with them), though I certainly do not expect other Americans to necessarily feel that way too. Many of these students in my lab know my personal feelings on the matter. But at the end of the day, there isn't much I can do right now to affect national policy, so I'm trying to think more locally. 

What, if anything, can I do here in my lab/department/university to help my fellow students/peers/friends?
The best thing you can do in my opinion is to be the change you wish to see in the world. Namely, be as friendly and welcoming to the international students as you can be. This will show them that negative views towards people who hail from the countries they come from are not shared by the entire US population, but are simply the opinions of certain people who happen to hold political office these days. It will also make you a role model for other Americans in your lab and department who may be inspired by your positive example to act in similar ways.