My guess would be that the sealer that holds the windshield in is bad and leaking. My car had this problem in 1997 and it was only 12 years old then, so imagine how much worse it could be on a car that is now pushing 31. The other source of possible leaks would be where the heater box joins the firewall, especially at the top of the joint under the inlet screen. I basically had to strip the windshield trough to bare metal, and then acid etch the rust out of it to fix it. I also had to fill in a pinhole or two with weld. I kind of know that my car has a death sentence with it's roof though, and will have to change it within the next 5 years. Even if I garage the car, it won't save it because of Florida's perpetual high humidity. The car that is in the garage now, and has not moved from there in 3 years, is now popping new little rust bubbles up in odd places. I figure I will replace the roof and floor the next time I paint it.
Also, you will probably need to replace most if not all of the seam sealer in the car. I did it on my car only to have it dry up and crack within 3 years. So, my solution was to use JB Quick epoxy this time. (I have owned my car since 1993) I also used it to fix a few pinholes in the trunk lip area after first etching them clean with acid. The drip rails on my car are all epoxy as I was tired of using caulk that didn't last. It wasn't cheap to do it, but I feel that it is a more permanent repair.
For right now, I would advice you to go to Wal Mart, get a big tarp (at least 10x10) and some bungee cords and cover the roof of the car, all the windows and the base of the hood by the windshield. This will keep the rain water out, and it shouldn't trap too much moisture if you work on it regularly. Get the heavy duty tarp sold in the camping good area. I think it is silver on one side and brown on the other. It will last for a few months unlike the cheaper ones which won't do you much good. Strap the bungee cords to the corners of the tarp and then fasten them down somewhere on the front of the fenders and back of the quarters to keep it taut. It will scratch the paint a bit, but I don't think that is much of a concern. You may also wish to either pull the drain plugs at the bottom of the back seat's foot well, or drill a few small holes to keep water from puddling for a long time.