first off, nice car. I love old bimmers; I got a '86 325 myself that i plan to drive until the wheels fall off, glue them back on, and drive it some more.

now, the battery light should mean that its not charging right or the battery is shot. i think auto zone or advanced auto can test both of those and charge the battery. thats a good place to start.
I also know that the e30s (like mine) have an issue where a single-shielded wire to the reverse lights runs past the shift linkage and can get worn through after time, causing a short. this drains the battery and just sucks in general.
so if its not the alternator, then it could either be a battery that won't hold a charge or a short. test for shorts by pulling the fuses and checking resistance across the circuits with an ohmmeter with the ignition off and the battery disconnected. some circuits should have a small leak current, like the clock and radio to maintain presets. but if his won't last a day, its probably a big leak or a crap battery.
best of luck!