The air pump is there to inject fresh air in to the exhaust system, ahead of the catalytic converter, to enhance operation of the catalytic converter under certain circumstances. The pump is part of a system called the Secondary Air Injection System.

The air pump comes on under the control of the car's computer (unless, of course, there is a problem with one of the parts of the system, as is in your case), via a relay mounted on the pump bracket. From they symptoms you describe, in all likelyhood, your relay is bad.

But, because you have moisture in the pump, it is likely that you have an additional problem. That additional problem involves the vacuum-operated valve that sits on top of the exhaust manifold at the rear of the engine. That valve is remaining open, when it should be closed, allowing moist exhaust gases to condense in the rubber tubing between the valve and the pump. And, because the pump is at the lowest part of this system, it fills with water, eventually destroying that very expensive pump.

The reason the valve is staying open can be traced to two problems. One problem is the valve itself may not be fully closing, perhaps due to a buildup of carbon deposits, or rust, or whatever. The second problem, and one that is commonly overlooked, is the vacuum solenoid, mounted on the top of the radiator shroud, has failed. It's the device, attached by a small vacuum tube, to the valve on the back of the exhaust manifold. When this vacuum solenoid fails, it allows manifold vacuum to pass, unregulated, to that valve sitting on the exhaust manifold. When this happens, that valve remains open all the time and your pump quickly fills with condensate.

Assuming your pump is still working, I would replace the relay, the valve on the exhaust manifold AND the vacuum solenoid on the radiator shroud. It is likely, also, that the little check valve between the vacuum solenoid and the intake manifold is also shot, so while you are there, replace that also. Also, replace all the rubber elbows in the system.