Many kinds of slate shower can be easily damaged by water. Slate shower is a metamorphic rock that was once clay and thus it can absorb water. If your slate shower is new you will want to skip to the last step and apply a sealer. If it is older and the waterproof sealer or tile sealer has worn off you can re-seal it by the following steps.
Clean the surface of the slate with shower cleaner and then wipe down with clean water. Get the surface as clean as possible.

Dry the slate shower completely with a towel. Fix any broken grout at this time.

If the waterproof tile sealer has completely worn off you will want to pull any water that has absorbed into the slate shower out. To do this place a space heater inside the shower stall. Use a heavy, 10 gauge extension cord no longer than 15 feet if you need more cord. Tape the faucet handles or shut off the supply to make sure nobody turns the water on with the heater inside.

Allow the heater to dry out the shower stall completely and to pull any moisture out of the tile. It may take 24 hours or longer.