In addition to statehouses, a surprisingly large number of governors’ races — as many as seven — are seen as up for grabs, including some that could alter control of capitals. If John Gregg, a Democrat, wins a tight race to replace Mike Pence as governor of Indiana, that state’s Republican-controlled Legislature will face roadblocks to its agenda. The same will be true in North Carolina if Attorney General Roy Cooper, another Democrat, wins.

In Missouri, a victory for Eric Greitens, the Republican nominee for governor, would end divided rule in Jefferson City, potentially easing Republican lawmakers’ efforts to pass measures such as “right to work” legislation. If Phil Scott, a Republican, becomes governor of Vermont, Democrats will no longer have total control of Montpelier and will most likely be derailed in their push for paid family leave and renewable energy development.

But the far larger number of state legislative contests could have a broader effect. At least 20 chambers are seen as highly competitive in the election, and a majority of those are now controlled by Republicans. Democrats are setting their sights on chambers not only in Minnesota but also in states like Maine, New Mexico and New York. Republicans hope to hold control and extend their reach in places like Iowa, Kentucky and Washington.

Such races are often decided by a few hundred votes or fewer, so turnout is essential — and, this year, a muddle. Democrats say their state candidates will benefit if Republicans appalled by Mr. Trump stay home. Republicans, though, say his core supporters are more motivated to vote than others, particularly given negative views of Hillary Clinton.

“In a lot of our areas where we are going on offense, he is extremely popular and Hillary Clinton is extremely unpopular,” said Jonathan Shell, a state representative from Kentucky, where Republicans have not held a majority in the House since the 1920s. Mr. Shell, a farmer who has worked to recruit Republican candidates, believes frustration over lost coal mining jobs and enthusiasm for Mr. Trump may finally help change that.

Democrats anticipate a different Trump effect. They have begun demanding that Republican opponents take stances on Mr. Trump and address his lewd remarks about women and allegations that he sexually assaulted women.