Dangers come in three main categories: slick surfaces, slippery spills and sliding objects.
First and sometimes easiest to deal with are sliding objects, such as toys left out, or dropped on the floor. Regular cleaning and a watchful eye can prevent these hazards from endangering players. Rugs and mats that slide over the floor can also be dangerous. One solution is to secure the rug to the floor, either with carpet tacks or double-sided tape. This will keep it in place fairly permanently, but this may not be what you want. You could try a skid-resistant adhesive or anti-slip bathtub stickers, attached to the bottom of the mat, a rough covering that grips the floor no matter where you put it down. The least permanent option is using non-slip rug pads. These will grip both the floor and the rug to keep it in place making moving and reuse much easier.
Slick floors can be a danger all on their own. Recently waxed floors, or any smooth surface combined with stockinged feet or platform heels can cause slipping and sliding for any player. This one is more a matter of careful planning and preparedness of the user. A recently waxed floor may look nice, but is probably not the best idea right before a play party. Everyone using the space should be aware of the floor and plan footwear appropriately. Slip-resistant soles or bare feet will generally be safer when the floor has a tendency to be slick. Just make sure the surface is clean enough for unprotected feet to avoid accidents.
Slippery spills can come from a lot of sources in a dungeon. Water, lube, bodily fluids, even wax play can create dangerous areas on the floor. If the furniture building is also done in the dungeon space, even leftover sawdust can create a hazard.
Keeping spills contained before they reach the floor is easier than cleaning up afterward. To do this a well-stocked dungeon might include puppy training pads, tarps, blankets, and towels. I have even seen a kiddie pool used for particularly messy scenes. Encourage guests to use these things, and remind them not to use liquids over the dungeon floor. Even spraying liquid onto something or someone can make the floor dangerous.
All the planning in the world cannot stop the inevitable. Eventually, things will spill, pads will be overwhelmed and splatter will miss the tarp. What then? A good cleaning kit can save the day. Mop and bucket, mild soap and water will clean up most messes. Be sure to have floor-specific cleaner to match whatever your dungeon floor is made of. For more oily spills like lubricant, you might need a degreaser such as acetone or a citrus product like Zep to break it up. Kitty litter is also good for absorbing especially thick spills. For wax on smooth floors, you can generally scrape it off once it cools, or iron it up onto brown paper if it is particularly stubborn or stuck to carpeted flooring.
All in all, careful planning, regular cleaning, and attention to detail can keep your floors, and your entire dungeon, a safe place to play.