Well, I don't think it's the heater core. You'd be smelling coolant inside the car, even more so when you turn on your fans. The aux water pump could be at fault. They can be had for about $50 if you can find them (will supply some of the part sources I use at the end).
http://forums.bimmerforums.com/forum....php?t=1158755
I wouldn't rule out the age of the piping, being an '89 and assuming it's never been replaced, the piping is 21 years old and probably pretty hard due to age and heat. I can't get a good look on my car (V12, no room), but most of the pipes that I can see are rubber. I'm would imagine they would be after all. Can you see if the leak is coming from a split in the tubing or a split in any metallic or plastic parts? If you can/want, pics might help.
If it is as simple as a split in the pipe work, you could simply call up BMW with the part number from the link above and order up a form fitted pipe, but a cheaper course of action would be to take the pipe off and inspect for any splits. If there are any, pop into your local parts store and buy a length of tubing. You can sort of eye ball the length you need when you get home, but you need to make sure the inner diameter of the new pipe is the same as the old one. Often they will sell in foot lengths, even if all you need is 3 inches you have to buy 12 inches, but that's okay, you'll have some left over for next time and a pair of large wire/side cutters or a sharp knife will be fine to cut to length.
If the pipes are fine on inspection, then I'd take the aux pump out and have a closer look and see what I can see.
Some parts sources:
http://www.autohausaz.com
http://www.bavauto.com <-- a little more expensive but OEM stuff.