The brief
The Jetta's coolant level had been dropping enough that the owner was topping up the expansion tank every fortnight, and a small puddle had started showing up under the front overnight. He brought it in before it climbed onto the temperature gauge.
Coolant doesn't go anywhere on its own, so a level that needs topping up every two weeks is leaking somewhere. The water pump pushes coolant around the engine and through the radiator, spinning on a shaft that runs through a seal, and when that seal wears, coolant weeps out past it a little every time the engine runs. That's the puddle. Lose enough and the level gets low enough to overheat the engine, and an engine running hot is one you don't want to keep driving.
The diagnosis
A pressure test traced the leak directly to the water pump shaft seal. The hoses, the radiator, the expansion tank and the thermostat all checked out clean.
That's a pump replacement, not a peripheral fix. The seal is part of the pump, and a pump that's started weeping only weeps more, so the unit gets changed.
The work
The cooling system was drained, the pump's drive belt section released, and the failed pump removed. A new VAG-spec water pump went on with a fresh seal, the belt set back up, and the system refilled with the correct coolant, the air bled out the proper way, and held under pressure to confirm the seals were dry.
A road test confirmed the level held and there was nothing dripping.
The outcome
No drips, the level holding cleanly, and the gauge steady through the road test.
The Jetta went home with the cooling system tight again. A weeping water pump only gets worse, and the failure at the end is an overheat that can cost a head gasket, so catching it on the first puddle kept it to a straightforward job.