Since you mention dept of water I assume you have municipal water service as opposed to a well with pump. But there is no pump in the municipal service.
Do you mean a sump pump? If so, its ground water and and the pump is pumping it out of your basement, as it should. Groundwater will most likely continually run to your pump, although you will notice seasonal changes based on the ground water table in your area.
If you mean something else, please provide more info, since I am surprised the plumber and water dept couldn't answer this