The fight over control of the House of Representatives runs through suburban battlegrounds, where Democrats and Republicans both believe they need to win the votes of college-educated white women, particularly independents who voted for President Trump in 2016. Those women are more inclined than most voters to say that Judge Brett M. Kavanaugh should not be confirmed for a seat on the United States Supreme Court following decades-old allegations of sexual assault against him.

Republican women running in those districts have had to calibrate their response to those allegations; to avoid alienating female voters, while not straying from the party’s support of Judge Kavanaugh, they have argued that his accusers must be heard. And in an indication of how fraught the situation is, many Democratic women running for those seats have so far largely held back from deploying the allegations as a weapon against their opponents.

Some Democrats have issued statements supporting Christine Blasey Ford, who has said Judge Kavanaugh sexually assaulted her at a drunken party when they were teenagers, and have urged the F.B.I. to investigate her allegations. Many have also supported another woman who says Judge Kavanaugh exposed himself to her during a drinking game when they were students at Yale.

Several candidates, including those running in swing districts in Pennsylvania and New Jersey, stood on stage at an event Monday for Emily’s List, the Democratic fund-raising group, wearing black in solidarity with victims of sexual assault and holding signs saying “I Believe Christine Blasey Ford.” Others embraced the #BelieveWomen hashtag on Twitter.