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It comes from the "Modesto Manifesto," which was never a written document but which Graham describes in his autobiography. Besides contact with women, it also involved resolutions about transparency in finances, the revivalists' interactions with local churches, and in publicity. It came out of discussions with other ministers about problems that plagued their line of work. Graham has a story about being seriously tempted while on the road, locking himself in a hotel room and throwing the key out the window. But he has also said that the "manifesto" was more of an informal agreement than a strict rule and that he's made exceptions as appropriate on various occasions. I think it was really composed with the particular concerns of being a charismatic public figure on the road frequently, not as a rule for life for all men.
You're paranoid.
Whatever its merits or lack thereof, invoking the Graham Rule as a reason not to be in a band with a woman is ridiculous. For one thing, it doesn't even apply. A band is not a one-on-one meeting. If you were to say you couldn't be in a duo with a female partner because you couldn't trust yourself not to make a move on her, or trust her not to accuse you, then it might apply. Billy Graham didn't refuse to have women on his team. All that invoking it this way does is to exclude women from the little boys' club of playing in bands, which is sad.
The thing about accusations is that, while false accusations do happen, there are far more actual incidents of assault and harassment that go unreported than there are false charges. Now everyone wants to go all backlash about how it isn't fair that after years of biting her tongue in terror of what people will say, a woman finally dares to tell the story of what someone did to her. What horrid is the way that so many are poised to jump straight to Potiphar's Wife references and slow to see Amnon and Tamar.
In the story about the woman and the trash collection, if she really had it in for this guy, do you suppose that not helping with her trash would really have protected him from her finding a way to make his life hell? Does it justify treating all women like crap because one of them might be psycho? Any dangers a man faces from predatory women pale by comparison to the dangers women face from predatory men. It's just a fact of living in a society filled with flawed people that you have to cope with the bad ones.
To be sure, there's common sense. I teach college. If a female student needs to meet with me, I don't invite her out for drinks or up to a hotel room. She comes by my office hours and I leave the door open. But then of course I don't invite male students up to hotel rooms either.
The problem with an awful lot of the posts on this thread is the immediate unspoken assumption that the #1 first thing to decide when contemplating a female bandmate is sex appeal - even if one eventually concludes "well, what really matters is her ability to play." To be sure, sex is a thing and it's not realistic to pretend that it's never an issue that plays a role. But that's true in all situations where people interact. The prospect of being in a band with a female musician has no more to do with sex appeal (as an asset or a danger) than hiring a female saleswoman to a company or studying under a female professor. Any hypothetical sexual interaction crosses the radar of your imagination, you assess it and figure out where the appropriate boundaries are, and carry on with what you're doing.