How do I keep drywall around a patch from crumbling?
Sounds like a bad spot in the drywall, maybe caused by water damage, or physical abuse. You're probably best to keep cutting until you find a good section, though you may get away with simply taping and mudding the joint. The tape should hold the section together, so even if the plaster is crumbling it will be held in place. If you opt to just tape and mud, there is no guarantee you won't develop a crack in the future.

If it were me, I would keep cutting (at least to the next joist). If you're lucky the bad section will not extend that far, or at the very least won't extend farther than a full sheet of drywall. You could try to patch it with compound, but it's not likely you'll be happy with the results.