Sal Oliva, a hotel worker and Uber courier from Staten Island, is ecstatic about President Trump’s executive order. So are Michael Bower, an owner of a home alarm company in Seattle, and Lynn and Don Broesch, retirees from suburban Milwaukee.

Mr. Trump’s new immigration policy may be setting off protests around the country and raising eyebrows and objections among allies abroad. But at home, a large portion of the electorate is behind the president. Mr. Trump’s supporters say that the promise of tougher immigration policies is one of the main reasons they voted for him. Friday’s order, along with an earlier one for building a wall, shows that he is serious, they say.

“I was like, ‘Oh, my God.’ I was so happy,” said Mr. Oliva, 32. He is gay and said he was deeply affected by the shooting at Pulse, a gay nightclub in Orlando, Fla., by an American of Afghan descent. “That one really got to me. That could have been me.”

Mr. Trump has tapped into a deep anxiety that is a relatively recent feature of modern American politics: terrorism from abroad.