“It does feel very much like 2010 reversed to me right now,” said Gov. Bill Haslam of Tennessee, the head of the Republican Governors Association. “There’s a lot more conviction about voting on the Democrat side than our side, which is a concern to us.”

While much of Washington was transfixed last week by the latest Trump-created uproar, this one over his widely criticized summit meeting with President Vladimir V. Putin of Russia, the nation’s governors gathered here for their annual summer conference to plot how best to exploit or mitigate a president who is as divisive as he is ubiquitous.

In a series of interviews, Republican and Democratic governors said opposition to Mr. Trump had galvanized liberal and many moderate voters, leading to a sizable intensity gap between the two parties.