New Jersey’s leading Democrats — including Mr. Booker, who has been spending much of his time out of state preparing for a widely expected presidential bid — are pleading for voters to rally behind Mr. Menendez. The senator’s campaign has sought to highlight the stakes by broadcasting a commercial portraying a vote for Mr. Hugin as a vote for Mr. Trump.

And most remarkably, Senate Majority PAC, a principal Democratic super PAC, has spent about $7 million on commercials in the expensive New York and Philadelphia television markets in a similar attempt to link Mr. Hugin to Mr. Trump. (The figure at the center of Mr. Menendez’s trial, Salomon Melgen, routed $600,000 through the same PAC to the senator in his 2012 re-election).

Allies of Mr. Menendez said his indictment and eventual trial made fund-raising difficult because nearly all of the money he raised was for his legal defense fund. And in the face of a self-funder like Mr. Hugin, Mr. Menendez had no choice but to rely on outside help.