I don't think OP would have posted this if they wanted to just leave it alone.
yes you have to cover it with something. tape would work (plastic, not paper, so it's water resistant), as would enough layers of paint, or a re-dip in the plastidip stuff that's already on it. The rust will get on your clothes if you use it long enough. It will also only get rustier, and will eventually literally fall apart
It wouldn't be worth my time to temporarily fix something that is obviously on its last legs.
I would get rid of it and either buy a new one or do what I do and put all your clothes on (plastic) hangers and dry hanging from the shower curtain rod. It's as little extra work as possible, since the dry clothes can then go straight to the closet.