I used to be one the side of "big athletics don't do much for a school." And then it happened to my school— I went to Gonzaga, and my senior year they did the NCAA March Madness blitz that took them to the Elite Eight and taught the nation to pronounce it properly. (Gone-ZAG-uh, as in "zig zag.")
The year after that, they took all of the dorms that had been converted to office space and changed them back to dorms. And they rented a wing of a hotel to use as freshman dorm space as well. When I visited the campus two years later, they'd put up a number of new buildings and were making improvements to the engineering and chemistry buildings. Now— a decade later— the footprint of the campus is roughly twice what it was, though the campus had owned much of the land before. (It had mostly been houses that the campus rented out; now there's more dorm space and other buildings.)
The population of campus isn't actually too much higher— about a third again, maybe 4000 undergrads. But the facilities improvement has been immense... and it's all because of the basketball team.
So— as close to a scientific study as you can get, before and after— it seems that a winning team CAN be really good for the rest of the school. I'm sure it can go the other way, but don't discount the power of sports.