The answer to this question is subjective and definitely open to debate. There are readers that are likely able to defend the idea that the Capitol controls the media to the point of brainwashing, but I would hesitate to say that the media is brainwashing people. When I think of media brainwashing, I think of it being more forcibly applied to people to create automatic responses and beliefs in a person. For a good example, check out A Clockwork Orange.
While I might be averse to saying that the media in Katniss's society outright brainwashes people, the media certainly is used to influence people by choosing what it shows audiences and what it does not show audiences. To me, that is not any different than present-day, real life media. For example, modern-day media is extremely good at creating stars. It's what shows like The Voice, America's Got Talent, and American Idol are all about. They show audiences key pieces of the "talent" in a way that makes us like and root for particular people. Key information is given to us that fits the image that producers and directors are trying to give audiences, and that is exactly what the Capitol does with the Hunger Games. Haymitch understands this, and he tries to educate Katniss on the power of the media. He knows that the media can make her look great, heroic, in love, and so on.
"You’re all they’re talking about. The star-crossed lovers from District Twelve!” says Haymitch.
“But we’re not star-crossed lovers!” I say.
Haymitch grabs my shoulders and pins me against the wall. “Who cares? It’s all a big show. It’s all how you’re perceived. The most I could say about you after your interview was that you were nice enough, although that in itself was a small miracle. Now I can say you’re a heartbreaker. Oh, oh, oh, how the boys back home fall longingly at your feet. Which do you think will get you more sponsors?”
If the correct images are given to audiences, then Katniss will have a better chance at survival because the media will help her gain sponsors. This is why I don't think brainwashing applies. It does not seem forceful enough to be considered brainwashing, in my opinion.