He would, after all, be only a year older than former Vice President Joseph R. Biden Jr., who has also made noises about running. And Mr. Trump, the oldest American to assume the presidency, will himself turn 74 in 2020.

But away from the boisterous People’s Summit in Chicago this weekend, some Sanders fans conceded their worries and suggested that the senator should focus on backing younger, fresh-faced candidates to push the Democratic Party leftward.

Max Weiss, a 19-year-old sophomore at the University of Illinois at Urbana-Champaign and the communications director for the Democratic Party chapter on campus, said he hoped Mr. Sanders would not run because it would be unhealthy for the party, which badly needs new faces.