@Ragman,
Once your batteris are charged (AND theres no trickle drainage from some internal wiring problems on your vehicles) they should HOLD a charge all winter. Is your cistern overfilling and draining into the sub bedding? That's really the only way that Id see a modern (stressed concrete) cistern from affecting the garage humidity.
Anyway, Id still keep a ceiling vent and dehumidifier in there to keep you from getting a mold problem down the road.
I had a golf cart at our place in MAine and I installed a breaker switch on the battery bank (I hd 4 batteries) so that , when we went home there was no possible draining from the golf cart.