As for the party’s emphasis on women’s issues, he said, “If Democrats weren’t running on these issues, the situation would be much worse.”

Image Cory Gardner, Mr. Udall’s Republican challenger, poses with a supporter. He won The Denver Post’s endorsement. Credit... Brennan Linsley/Associated Press

“The headwinds that you get from Obama and other factors affect everybody — they don’t only blow in the faces of men,” Mr. Garin added. “Even in the face of those headwinds, Democrats are still much better able to succeed with women voters than with men voters.”

In Kentucky and Louisiana, new polls grabbed attention for suggesting that Republican Senate candidates had made inroads with women. Senator Mitch McConnell of Kentucky, the Senate’s Republican leader, was essentially tied among women with the Democratic challenger, Alison Lundergan Grimes, in a Bluegrass Poll taken Oct. 25-29. But the poll, by SurveyUSA, does not meet polling standards of The New York Times because it was partly conducted using automated phone calls.

Louisiana’s embattled Democratic senator, Mary Landrieu, narrowly led her Republican challenger, Bill Cassidy, among women and men likely to vote by Tuesday, according to a poll conducted Oct. 11-24 by the University of New Orleans Survey Research Center. But both groups flipped in Mr. Cassidy’s favor for the expected December runoff election that will be required by state law, assuming neither candidate exceeds 50 percent.

Yet in a number of battleground states for the Senate and for governor, Democrats continue to have enough of an advantage among women to be in contention, even though many of the states are heavily Republican.