If there was a physical connection between Lake Seneca and the salt down, the salt dome would be dissolved until the lake reached maximum salt saturation. It would be far saltier than the dead sea.
We use water to create the caverns in the salt dome. Once the water nears salt saturation the water has to be replaced to keep growing them. Handling the fresh and the brine water is a major portion of the operations at the facility I work at.
It’s more like a slow seepage from below. The lake is being constantly supplied with fresh water from the runoff from the surrounding hills. There are even numerous water falls leading into it.