I screwed up one day and ran a dozen .38 special brass, primed, through my ultra sonic cleaner for 16 minutes. When I discovered it I sat them up on a shelf in my reloading room and let them dry for a month. They fired just fine. Couldn't tell them from new primers.
The point is that primers are pretty resilient and are hard to damage with moisture.
You could store them in an open 5 gallon bucket if you kept them in an air conditioned area, and never have to worry about humidity hurting them.
I'm not sure humidity could hurt them anyways.