It isn't an alternator, the only real reason a vehicle has a battery is to start the engine. Once the engine is running the alternator powers everything. If the engine dies going down the road and all electrical power is lost (headlights, radio ......) it is a charging system problem. If the engine runs fine but has to be jumped to start it is a battery problem.
That doesn't mean the battery itself, as
@Nex has suggested twice do a draw test. I bet you have something not turning off and draining the battery. Ironically I have see a few alternators cause this (pretty rare, I am drawing from 35 years experience) if regulator shorts and doesn't shut off when the engine shuts off.
two things I like to do. disconnect the negative battery cable and let the vehicle sit as long as possible, minimum 1 day but 2 days if you can. hook the cable back up and see if the engine will start and/or if you have electrical power. If you do then the problem is not the battery and you have a draw (they can be a bitch to find, especially if they are intermittent). If after hooking up the negative battery cable you have no electrical power then the battery is shorted internally and drawing itself down.
other thing is, let the vehicle sit a few hours. long enough for the engine to cool down to ambient temp. then open the hood and put my hands on various components feeling for warmth or heat. Most specifically the alternator, fuse panel in the engine bay and inside the cab.
I don't know of anything specific on a gen 1 that causes a draw.