WASHINGTON—Steve Bullock, the governor of Montana, entered the crowded Democratic presidential contest Tuesday morning with a strong emphasis on his record running a red state.

Mr. Bullock, a second-term governor, is the 22nd major candidate looking to challenge President Trump as the Democratic nominee next year. Key to his pitch: He is the only Democratic hopeful elected to statewide office in a state Mr. Trump carried in 2016.

In...