WASHINGTON—Donald Trump’s tumbling poll numbers have given Democrats their strongest tailwinds in a decade, heightening expectations for significant gains in the House, though likely not enough in the eyes of most analysts to shift control of the chamber.

Republicans have sought to contain the downballot damage, with some distancing themselves from the top of the ticket completely after a 2005 video emerged in which Mr. Trump made lewd sexual remarks and women said he touched them inappropriately. Speaker Paul Ryan (R., Wis.),...