the rubber seals of the water pump are not in contact with coolant, the seals in contact with coolant are ceramic and rubber is behind them.
If your system is heavily rusted - clean it out with citric acid flush (use search) you MAY or MAY NOT have leaks afterwards depending on how bad your system is rusted. Once clean flush it out about 4 times with clean water.
DO NOT USE COLD WATER TO FLUSH OUT THE SYSTEM ON A HOT ENGINE. It will cause damage.
do this with no thermostat valve to make sure the pump flows full force through everything.
Install the thermostat valve back after clean water flush and purge from the block too - use air to purge everything before installing the thermostat valve - fill with prepared coolant, bleed out air bubbles, ensure the cap is good, ensure the overflow tank is good, ensure the overflow pipe nipple on the radiator and the overflow bottle is good (leaks here can cause the system to go low as air can leak into the system when it cools down)