That is actually a feature on some boats. The well can also fill through the drain depending on whether it is a low or a high one. I suppose it could be water forcing it's way through the livewell pump but I am not sure. To see if it is the drain that is letting water in, install the overflow tube or put a cork in the drain (if located at bottom of tank).
If it is coming in through the fill fitting and through the pump, first check to see if there is a valve on the fill fitting in the livewell. If there is and never was a valve, you will have to close the seacock. 'My other boat' has a fill fitting that doesn't look like a valve but is and can be turned to regulate the flow. On that boat the flow rate.