Leaking Water Pump
by Ken Streeter, with input from various sources
If the water pump is starting to go bad, there is a weep hole on the underside of the water pump, that will let coolant drip out, as an early warning signal that your pump might be bad. (It is intended to let excess lubricant from the bearings drip out rather than get forced past the seal into the coolant, and also let coolant drip out when the seal goes bad rather than contaminate the lubricant in the bearings. )
If coolant is leaking from this hole, the water pump is starting to show signs of wear. In some cases, the pump will slowly weep on the first few runs after sitting for a while (months) and then stop. If your car is a daily driver and its weeping, then I suggest replacing first off. If you just brought it out of winter storage, watch your fluid levels, engine temp, and the rate of the leak for about a week of running. If it doesn't stop, then replace. Don't go any longer, the pump might sieze or fail!!!
An additional check for water pump condition is to remove the belt and jiggle the rotor; too much radial play points to a bad bearing. New water pumps are not too expensive, and fairly easy to install. It is generally simpler to buy a new (or rebuilt) pump assembly than to try to repair an existing pump.