I store them dry and wet it with hot water right before a diaper change. The bathroom is right next door to the nursery, so it's easy, but I don't like the idea of damp cloths sitting around all day anyway.
Ditto the wipes in a basket and dampen them with warm tap water just prior to use. You can find recipes online for wipes solutions, and there are videos on Youtube for how to fold them so they come out of a disposable wipes container. My experience is that they get moldy faster than I'd like, especially in a wipes warmer.
Re: wipes