P bass pickups are about the most dirt simple thing in the universe and they're very easy to diagnose.
Really, the only possibilities are that it could be a short in the coil, or it could be a problem with the magnet. A coil short across many windings can result in the pickup acting as if it had that many fewer windings, which would recude output. Alnico magnets are easy to unintentionally demagnetize, which can result in low (or no) output.
Do you have a multimeter? Check resistance of each coil. Multimeter in DC resistance mode, one probe on each eyelet of the coil. If the coils read different resistance values, one of them has a problem (presumably the one that sounds bad). If they're the same (or within, say, 10%) then the coils are likely fine. Which leaves the magnets.
Most people don't have tools to measure magnet strength. You can sorta check magnets just by holding a paperclip or something else magnetic nearby and try to gauge the pull. If the weaker pickup feels like it's pulling less, the magnets may have been demagnetized. Or if you don't want to bother checking, you can just assume it's a problem and fix it, which is really easy to do, and can't harm anything (if the magnets are already charged and you try to charge them, it doesn't change anything). Here's the man Lindy Fralin showing how it's done:
