We just bought an older home that had new perimeter drainage system installed and a new sump pump installed that drains outside. The old sump was closed off and now there is signs of water leaching up mainly along the line from the old sump to the floor drain. There is crumbling cement, white powdery substance staining the floor and today actually in a few spots there is a white fuzzy substance appearing. Other cracks in the floor are now show signs of the leaching also. I am thinking the old sump is filling up and has no where to drain which is causing the problem.
It may be the case that the old sump is filling up. I'd suggest opening up the floor and investigating as it's the only why to find out what's happening and fix the problem.
You could coat the floor with a water repellent sealer. This may stop the water from coming through the floor, but would not fix the underlying problem.
Also, check to make sure no household drains or exterior downspouts are being piped to the old sump location. This is easier said than done, but something to consider.
Otherwise, the white powdery substance is most likely efflorescence. This is a condition where salts are being moved out of the concrete due to the moisture moving through it. It's harmless to you but shows signs of a problem in the floor (likely due to the water).