How can i stop the water that splashes under my clawfoot tub from leaking into the apartment below me?

I just moved in to a new place that has a clawfoot tub. Everytime I shower water gets on the floor under the tub, and leaks into the unit below me. The tiles under the tub need to be replaced. They are very old and the grout between them appears to be rotting, and has holes in it), but will this alone solve the problem? Someone also recommended installing a drain under the tub. Do the walls around the tub need to be tiled too? The “handyman” that services the house does not have my trust. He wants to replace the tub with a shower stall (not happy about that idea AT ALL) or build an enclosure around the clawfoot tub so that it more resembles a standard tub. Any advice on this matter would be so appreciated. HELP? thanks!