There are numerous threads on parasitic drains leading to dead batteries during storage. We had this problem, too; and it was resolved by installing a Perko marine battery isolator, which completely disconnects the batteries from all trailer connections. This can also prevent the other common problem, overcharging batteries when connected to shore power 24/7, by removing the batteries from the charging circuit. Plus, all electrical devices and electronics, including the refrigerator, furnace, water heater and propane detector, are still operable using only converter power.
We have two Blue Top Optimas, which will now retain a charge for several months with the battery switch set to OFF. Optimas are supposed to hold a charge for up to a year, although I can't imagine leaving our Airstream sitting unused for that length of time.
For more information, search for "Perko" and "Battery Isolator Switch" on these forums. We installed a Perko switch, because of previous experience with it on our boat. However, there may be other brands and devices that will work as well, or better.