ARNOLD, Mo.—Sen. Claire McCaskill stood in front of a group of about 25 voters and volunteers in her campaign office here last week and told them her re-election race could come down to the number of people in the room.

Ms. McCaskill, the sole statewide Democrat in this increasingly conservative state, faces her toughest fight yet to be re-elected for a third term. She is facing Republican Attorney General Josh Hawley, who is tying himself to President Trump, who won the state in 2016 by more than 18 percentage points. Polls…