About 60 percent of voters said trade with other countries caused job losses in the United States, according to the latest New York Times/CBS News poll. Mr. Trump has a slight edge over Mrs. Clinton on the question of who voters consider better poised to tackle the trade issue.

Putting Mr. Kaine on the ticket “could be disastrous for our efforts to defeat Donald Trump in the fall” because of his support of the Trans-Pacific Partnership, said Charles Chamberlain, executive director of Democracy for America, a liberal political action committee.

Mr. Kaine, the son of a welder who owned a small metalworking shop in a Kansas City suburb, could help Mrs. Clinton attract white male voters and independents. Those voters may prove more critical to her chances in November than the young liberal voters who backed Mr. Sanders in the primaries.

As governor of Virginia, Mr. Kaine appealed to both Democrats in urban pockets and independents in rural areas, and established a reputation as a pragmatic consensus builder. “He’d appeal to people in the Midwest because that’s his roots,” said Carl Tobias, a law professor at the University of Richmond. And Mr. Tobias said Virginians across political lines considered him “a thoroughly decent and honest person.”