The windshield gasket is probably bad and you might have rust.
+1 on the nuts diagnosis. If there is a leak in the windshield gasket, it will most likely be rotting the window frame (rust). Rain will get in and present itself as it drips from inside the front pillar. Get in there with a headlamp and look for water residue. The real giveaway is if there is a crack in the window at the edge on the side where it leaks. The weakening frame no longer supports the glass in that spot and it will eventually crack.
To check, the damage it is an easy task to pull the rubbery locking bead from the center of the windshield gasket.
HOT TIP before you do: take a sharpie that will not be absorbed by the rubber and mark every 6 inches across the rope and the gasket, and also where the ends began and end. When you put it back in the marks will help you replace it without stretching it or running out before you reseat the bead.
CARNAC says it could be heater core related- yes, on that one too. Also check to see if anyone (the PO) has installed lights or antennas and left holes open on the roof.
Either way stop the rain from getting in NOW! Check under the floor mats to see how long this has been a problem. Air/tear it out if the pad is soaked. Treat it with your choice of sealant and lay new pads in. Don't let the leak get away from you.