On campuses clenched by unforgiving debates over language and inclusion, some students embrace Mr. Trump as a way of rebelling against the intricate rules surrounding privilege and microaggression, and provoking the keepers of those rules.

Among older whites unsettled by new Spanish-speaking neighbors, or suspicious of the faith claimed by their country’s most bitter enemies, his name is a call to arms.

On the internet, Mr. Trump is invoked by anonymous followers brandishing stark expressions of hate and anti-Semitism, surprisingly amplified this month when Mr. Trump tweeted a graphic depicting Hillary Clinton’s face with piles of cash and a six-pointed star that many viewed as a Star of David.

“I think what we really find troubling is the mainstreaming of these really offensive ideas,” said Jonathan Greenblatt, the national director of the Anti-Defamation League, which tracks hate groups. “It’s allowed some of the worst ideas into the public conversation in ways we haven’t seen anything like in recent memory.”