HELENA, Ala. — It was unusual enough for Keith Dorsey to open his door in this heavily Republican Birmingham suburb and be greeted by a Democratic canvasser, a sight normally as rare here as a Clemson fan. But these are unusual times.

“This is our only opportunity,” Mr. Dorsey, a Democratic-leaning engineer, said, referring to the tight Senate contest between the increasingly embattled Republican Roy S. Moore and the Democrat Doug Jones. “We need to seize it.”

Even the most optimistic Democrat knows it’s not at all clear the party can.

Mr. Moore, long a controversial and polarizing figure, stands accused of molesting or making unwanted advances toward numerous young women and girls, one as young as 14, when he was in his 30s. That has rendered him radioactive for national Republicans and led The Birmingham News to bellow on Sunday’s front page: “Stand for Decency, Reject Roy Moore.”

And the Democratic candidate, Mr. Jones, is a respected former prosecutor best known for convicting two Ku Klux Klan members for the 1963 bombing of the 16th Street Baptist Church in Birmingham, who was raising the possibility of a truly competitive race even before the accusations against Mr. Moore.