As Eldyfig suggested, the seal can start leaking between the drive pulley and actual pump. If that's leaking, the solution is to replace the pump cartridge. You'll need the tool shown above. The new pump is AC-Delco part number 252707 for your car, and it costs about $55-$75 depending on where you go. I would look on
www.gmpartsdirect.com or
www.rockauto.com for a good price. The new pump will come with a rubber O-ring seal that goes behind the pump, but it does not come with the water pump cover gasket, and I would recommend buying that separately and replacing it at the same time.
When you do this, make sure to clean off all of the gasket sealing surfaces very thoroughly with something that won't damage the finished surface. (plastic putty knife, maybe) Some crud can accumulate in some of those areas, and you should clean it all up well before you put it back together. No need to be taking the thing apart again to fix a leaking gasket caused by crud that didn't get cleaned up while you were in there.
Refilling the system, be sure to use Dex-Cool coolant (orange stuff) and get 2 tubes of Bar's Leaks Golden Seal powder or 6 Bar's Leaks or GM coolant supplement pellets. If you get the pellets, put them in the radiator hose, DO NOT PUT THEM IN THE SURGE TANK!!! If you get the powdered stuff, mix it in with the first gallon of coolant and shake well before dumping it in the surge tank. Then dump in the second gallon to push all the supplement powder through the system. If you just dump it in the surge tank, it will just sit there as the surge tank has almost zero coolant flow through it. Mix up the coolant 50/50 with distilled water and refill.