Power
Not so much. The guys at the top of their game (ha) like Costas, Buck, and Michaels have pretty solid contract negotiating power, but that's about it. Recently Costas tried to flex his societal influence, testing the waters with some public anti-gun rhetoric.
It didn't work out too well for him, though. He didn't lose anything for trying, but the general response was, "Go back to sports! If we wanted to hear about politics we wouldn't be watching football!"