Lots of layers of newspaper or cardboard can help (with something not-slippery on top of that.) The layers will keep the chicks from directly scratching the bottom of the box, or pooping directly on it.
Keeping it dry can help keep the bottom in usable shape. Maybe put the water on a plate or a baking sheet or something of the sort (ideally, something you can throw away when you are done with it: maybe the lid from some kind of food product.)
The bottom just will fall apart at some point. If you have it sitting on something waterproof and semi-sturdy, you will be able to pick up the whole mess at the end and carry it out to wherever you dispose of it (probably trash can or compost pile). The "something waterproof" could be a tarp, a plastic shower curtain, a big plastic trash bag, even the plastic bag that some brands of chicken food come in. I've used a variety of things, depending on what I already had (especially things that were going to be thrown away anyway, that I could use and then toss!)
My heating pad cuts out at least once every time I brood chicks, but I've never lost a chick from it. They just hunker down and huddle up until I get it back on.
