Standing by and tutting as everything goes to hell is not good enough. You have to get involved, you have a duty to help to try to put things right in some way. By doing nothing you’re implicitly siding with the offenders, your apathy lending weight and credence to their cause.
That’s the idea. But then, being a human idea, it’s exploited and twisted as much as it's employed by those with good intentions. It’s used as cheap justification and easy emotional blackmailer when trying to drum up support for just about any cause – even the outwardly virtuous ones. Take environmentalists. Don’t recycle? You’re part of the problem. Don’t switch your TV off standby? You’re part of the problem. Don’t drive an electric car? Part of the problem. Don’t live in a cave, eat carbon-neutral biodynamically grown vegetables and shit in a composting toilet? You, my friend, are not part of the cure, you’re part of the problem. Because of you, the planet is dying.
Meanwhile, certain right-wing religious groups in America use the slogan as justification for naked bigotry: according to them, AIDS is a disease sent by God to punish sinners. And if you don’t contribute to their hate-filled creed, if you don’t become a part of their “cure” by targeting homosexuals yourself... well then you’re part of the problem.
We’re all part of the problem, ultimately. The best we can do is limit that part to a walk-on.