Natural cleaners. Virtually all commercial cleaners contain chemicals that are toxic. Vinegar (white, distilled), diluted 1:1 with water, makes a great general purpose cleaner. It cuts soap scum, removes mildew. Note that it can also dissolve tile grout or discolor some materials, so try it out first on an inconspicuous area. Search "better living through less chemistry" for more.
Use baby's-eye-view. Before a baby crawls at home, a parent can crawl around to look for baby hazards or chokable items. Be sure all hanging things (such as tablecloths, electrical cords, drapery cords, etc.) are secured up high. Be sure the furniture has no sharp points that can poke a baby going underneath furniture.
Green cleaning. You can always use water and soap, or even add some lemon juice and/or baking soda for added cleaning help. Additionally, there are numerous green cleaners now available that avoid toxic chemicals and are safer for crawling babies.
Dust mop. Depending on how dirty your floors get (with three kids, mine are usually filthy within minutes of cleaning), a dust mop or broom should be fine for everyday cleaning. For sticky spots, there are lots of organic cleaners on the market or just use hot water and elbow grease. For carpeted floors vacuuming without a special cleanser is fine.
Soap and water. Any commercially available floor soap and water will be fine.
Natural cleaners. Agree that vinegar and water is actually a very good cleaner and sterilizer. Works as a great carpet cleaner too. Also, there are plenty of natural cleaning products -i use babyganics (no personal or financial connection for me) at my house for everything from floors to hands to pacy cleaning - but they are expensive. Soap and water plain works great too and is the original anti-biotic cleaner.
Dry floor . Any regular household cleaners should be fine as long as floors are dry when the baby is on the floor. Make sure to lock your cleaners so the baby could not get to them.