“Some of these sentiments were beginning to percolate, but it was the task of political leadership to keep the ugliest elements in check,” Mr. Wehner said, offering a damning conclusion about Republicans today versus then: “It was a very different party.”

Mr. Trump’s hyperbolic, confrontational approach to taking on opponents — Hollywood stars, television news anchors, professional athletes and just about anyone else who offends him — has left as much of an impact on modern politics as his policies, if not more, strategists said.

He set the tone early, opening his presidential campaign in 2015 with the startling declaration that Mexico was sending its rapists and criminals rushing across the southern border.

Some Republicans say following the president’s lead will only hurt the party’s candidates.

“Everything is being seen through the filter of Trump,” said Russ Schriefer, a Republican consultant who is working with Gov. Larry Hogan of Maryland, whose Democratic opponent, Ben Jealous, is black. Mr. Hogan has been one of the rare Republicans to successfully create a groove for himself in the party that is entirely distinct from Mr. Trump.

When you try to act too much like Mr. Trump, Mr. Schriefer added, “You’ve come to the consciousness of the voters based on that, so it’s a lot harder to escape.”

Many candidates, however, see little reason to want to escape Mr. Trump’s shadow given his overwhelming popularity with Republican voters — which Gallup measured at 85 percent in its most recent poll.

And many of the issues that involve race and ethnicity are part of the party’s core messages on subjects like immigration. In many ads, Republicans are invoking the Latin American MS-13 gang as a growing menace. They are linking Democrats to the left-wing movement to “Abolish ICE,” the Immigration and Customs Enforcement agency.