I'm no expert on this but I guess you're thinking that since Red Cedar grows well in damp, moist climates, then it must be OK in a damp basement. Well, unfortunately, I think you're streatching it.
Above, I'm talking about live cedar. You're talking about 'dead' cedar. Live cedar has bark and is outside and has sun and so on. Not your closet. If it helps you decide, unless your basement is dry, I wouldn't store clothes down there. Ours is dry and we DO store clothes down there. IF your basement is heated and cooled as any other room in the house, then I say it's OK to do as you describe with EITHER closet. But if the basement is damp AND there's a way for moisture to rise up and get 'trapped' (such as in a closet), then you may be asking for trouble. Mold and such don't need sun to grow. It needs moisture, 70+ degrees, and a surface to attach to. Altering ANY 1 condition just enough may prevent the mold from growing. Just look at your bathroom(s) and there's a good chance you'll see mold in the shower or up in the corners. (It may be just slight but it may be there.)
As with most wood, cedar will expand and contract so if you're building one, be sure you 'make it' in such a way it can accommodate that. Also, cedar closets and such are used to keep out moths and most 'every day' insects.