Your Alternator charges you battery has nothing to do with your ABS ( Automatic Braking System)
there could be and electrical short in your system some where or a bad ABS sensor, thats causeing feed back through system shorting out your alternator ,it would be a trail and error project for who ever works on it
I think i would unplug my ABS and try that ,your brakes will still function , you will just slide your tires if you have to stop quick , you just wont have that grab and release you have with ABS
A car that old could have a lot of things wrong with it might be time to trade it in
At any rate i would get a lifetime warrenty on the next alternator i bought


