HOUSTON—The third Democratic presidential debate left the candidates differing on whether it is fair game to discuss former Vice President Joe Biden’s fitness to be the party’s presidential nominee next year.

Mr. Biden has maintained his front-runner status in public polling despite a series of verbal missteps during the summer. Some Democrats have warned that his misstatements and gaffes could undermine Mr. Biden’s case that he is the one best placed to beat President Trump. And at least one of his lesser-known Democratic...