A daily hoof picking is a good idea no matter what conditions the horse is standing in. Even babies can pick up stones or stand in poo in the pasture, and hoof picking is a great time for you to inspect the foal's feet and legs for scrapes or heat, and treat accordingly.
There ought to be some kind of bedding in the stalls, regardless of what the bare flooring is. What kind is up to your personal preference and regional availability. I've heard everything from straw to shredded newspaper and peatmoss. I personally use sawdust or bagged pine shavings if none of the local sawmills are selling to individual buyers.
Not only does the bedding soak up waste, but it provides a comfy place for your horse to bed down in a box stall. They don't lay down much in a day, but when they do the definitely appreciate a little something to cushion the floor.
A stubborn horse walks behind you, an impatient one in front of you, but a noble companion walks beside you ~ Unknown