EDGARTOWN, Mass.—One of Vice President Joe Biden’s top challenges, amid increasing signs that he may announce a presidential campaign, is determining whether he could corner a Democratic constituency that would sustain his candidacy and make him a formidable threat to Hillary Clinton.

With Mrs. Clinton slipping in polls, Mr. Biden could introduce another challenge to her campaign by drawing away key voters: working-class Americans and independents. Both groups have moved away from the party in recent elections, and Mr. Biden...