This happened to me last week and i have the original windscreen, with Japan at the top of the screen.
I discovered the rubber, at the drivers side bottom corner, had perished, bits had come away and their was a lot of cr4p gathered under the rubber moulding. I did excactly what true blue said, then bought some windscreen bonding stuff from Halfrauds, cleaned out all the gung, and sealed the area that had perished..
Iv'e had the car washed and it's been pouring down since, and it's still ok.
Joneb- i'm not very mechanically minded but this is NOT hard to do mate. Just take your time. You only need to take off the cover that surrounds your fuse box There are 3 screws under the steering column which takes the under cover off, then where your accelerator pedal is, their is a big plastic bolt that should just screw off by hand. Once you start, you'll see how to get it off. It may not seem wet or damp but this is defo your problem. Just disconnect each connector and give them a wipe, i got into them with cotton buds, then get the hair dryer onto them. JUST DO IT MATE you'll be fine.
Dont forget to check round about your screen to see where the water is coming in, if you dont see to this then it will just keep happening.
Hope this helps and good luck.