Lousy cheap plastic dropcloth material won't hold up. It not only will be getting tread on by x number of campers and y number of visitors all week (who will be wearing shoes/boots/etc), but it will be subjected to winds of both the crazy and non-crazy variety. As AntiM suggested, you want something sturdy.
In my opinion (and experience), it's not necessary to go 'wall to wall' with flooring under your shade. If it's a well designed (and executed) structure there won't be a whole hell of a lot of wind in there. Strategic bits of flooring are the way to go. I recommend to our campmates to either get carpet scraps, a small rug, or even just a welcome mat to put at the entrance of their tent (or in the case of my gf and I, outside our van). That offers a place to take shoes off and put 'em on when going into or out of the tent/van/what-have-you. This greatly reduces the amount of dust that gets tracked into private spaces, as long as a person uses something like this and then actually remembers to keep their tent zipped up tight all the time they won't have any trouble with dust where they sleep. From there, we have a couple decent sized rugs we put down in the common seating area of our camp's private space. It's a creature comfort, for those who enjoy "lounging around the house" in bare feet.
On a related note, I'll highly recommend that you guys consider bringing fixins for playa foot salad. Quite possibly one of the most awesome tips I ever got. It's easy to do. Get (for the whole camp) a cheap wash basin (the kind used for washing dishes works great), a washcloth, and then bring a bottle of cider vinegar and a bottle of olive oil. Every day or two, or any time that a campmate's hands or feet are starting to feel a sting or a little rough around the edges, it's time for playa foot salad. Douse the washcloth with vinegar, and then wash your hands and feet with it (use the basin so nothing hits the playa). Next, massage your hands and feet using a little of the olive oil. The vinegar is slightly acidic and counters the alkali (base) of the dust, and the olive oil both takes the edge off the strong vinegar smell and serves as a great moisturizer treatment. You'll smell a bit like a salad for the next few minutes (hence the name), but your hands and feet will feel great.