It is possible to attract favorable media attention, especially locally. A few years ago, our club's Scholastic Clay Target Program team won the state SCTP trapshooting championship. I was helping coach the kids at the time and was there, So I immediately sat down at my Blackberry and composed a press release, tracked down the email addresses for the sports editors of a couple of our local papers, emailed the press release and included my cell phone number.
A few hours later I got a call from a feature writer for one of the local newspaper chains, and I arranged for her and a photographer to come out to the club the next week to interview the kids. And that resulted in a very nice article being published in all the chain's newspapers, covering roughly the entire East Bay side of the San Francisco Bay Area.
See
http://findarticles.com/p/articles/mi_qn4176/is_/ai_n15831214 (Unfortunately, the version of the article in this link doesn't include the photographs).
A little while later, the anchorwoman of one of our local morning news shows challenged the woman reporter on that station's evening magazine show to a shooting contest. Our club was contacted, and we arranged for them to shoot some trap. We also provided some coaching for the anchorwoman. We got a nice segment on the show, and the two reporters had a good time and came away with a positive view of shooting sports.
Sure these are only local media, but I think things start there. If you are active in a club, develop lines of communication with your local TV stations and newspapers, and keep them informed of activities at your club.
And yes, in both cases the focus was on the sporting aspects of guns and shooting. But that is often the most effective way to introduce the subject to the uninitiated and begin to dispel the "demons" many people associate with guns. IME, once someone has begun to see guns in a positive light through the shooting sports, they become more open to rational consideration of self defense applications.