How many batteries have you been through, and are you sure they were fresh and in good shape?
Do you own a multimeter? Can you check the wiring on the jack?
Active basses typically use a stereo jack as a "disconnect" to turn the battery off when the bass is not in use. A stereo jack has a separate connection for the ring and sleeve, but a mono cable (as you'd normally use for a bass) just has a really long sleeve instead of a separate sleeve and ring. Your jack should be wired with the ground for the preamp and other electronics connected to the sleeve terminal, and the black wire from the battery connected to the ring terminal. Thus, when the cable is not in the jack, the connection is broken, and the battery is disconnected (and doesn't drain). But when you insert a cable, the sleeve on the cable connects the sleeve and ring terminals, which grounds the battery, and the electronics receive power.
Problems like you're describing are often caused by an issue at the jack. Sometimes the terminals will be bent or the jack will be poorly aligned, and there will be a short. For instance, if the cavity is shielded, and the ring terminal (where the black wire from the battery is soldered) is touching the shielding, that will basically bypass the switching action and leave the battery connected 100% of the time. Other times, an ignorant prior owner may replace the stereo jack with a mono jack and not realize that they're defeating the switching action. Or some other problem will leave the battery connected 24/7.
So, pop the cover off and have a look at the jack as a starting point. Show us some photos if you're confused.
