WASHINGTON—Elizabeth Warren is building a broad coalition of Democratic voters, and her rivals are taking notice. But so far, they are still testing out ways to undercut her appeal.

Ms. Warren’s steady rise has prompted primary opponents such as Bernie Sanders, Kamala Harris and Pete Buttigieg to search for ways to blunt her momentum at a time when the leading contender, Joe Biden, has weathered a series of stumbles and direct attacks.

While...