you can control the electric windows through the vag diagnostic software. if you take the ashtray out and slide the trim to the left of the ashtray to the right (think that makes sense) you will find the port. this can tell you any faults and re-programme things like the windows opening and shutting with the key in the door etc. of course it could be something as simple as a bad earth, but the mk3 golf window mechs are notoriously dodgy for such things as cables jamming and the motors wearing out. i would suggest having the door card out and checking that it all looks and works as it should. if nothing obvious try the vag software for faults.
as it happens i have had to replace 2 window mechs. i have always found replacements on ebay for £25-£40. it is quite an easy job just make sure you dont drop the glass
