How many folks write $$$$$ checks to the science department? How many fans buy tickets to watch a history lecture?
Taxpayers have much to gain when their state schools are playing well. The revenue sports deposit not withdraw from the university and, perhaps more importantly, inject cash into the local economies.
Locally here UT-SA has seen a nice increase in applications from out of the area students. They attribute it in part to the Roadrunners football team being on national TV eight times last year and playing in a bowl game. It put their campus in front of more kids than an ad on Twitter or a splashy website.
I teach seniors in a class that helps them get into college, pay for college, and be successful while in college. I will have 20 to 30 recruiters in my classroom throughout this fall. Every one will either brag about how much fun attending big games are, they will highlight some sports centric activity, or if they don't have athletics they will try to minimize the factor. It's the front window to the university.