WASHINGTON—Larry Cohen, a top adviser to Sen. Bernie Sanders, is charged with winning support from labor union leaders. Lately, it’s been an uphill climb.

On Tuesday, the mammoth Service Employees International Union endorsed Democratic presidential rival Hillary Clinton, her 13th national union nod compared with Mr. Sanders’s two. Mr. Cohen hopes to expand this number, but absent that, would be happy to keep unions on the sidelines while Mr. Sanders makes his case.

Mrs....