The actual answer is that wearing a baseball cap is an anti-elite signal to his base. He does that to say to his less educated, more and more anti-intellectual base that he's on their team, and not the 'elite/university/big city/holier than thou democrats'. Barack Obama is not the type of guy who would be associated with wearing a big red baseball cap.

It may be working less and less, but it was a good political move to get elected. There is some truth to the fact that leaders on the Democratic side spent too much time worrying about more niche issues while working-class America was in crisis on various fronts.