I have the same problem. First, to help prevent it I loop the bottom part of the curtain through the hose for the handheld showerhead for a while to let it dry a bit. Then I pull it down and let it hang outside the tub. Helps a lot. When it eventually gets a bit moldy I soak the bottom in bleach. Tried Oxyclean and other things but only bleach worked. Then I rinse it and wash it in the machine with the bath rugs. Comes out very clean.

I just use a white hookless curtain made of polyester - I don't use a second curtain. Found that I had more mold with the second layer of curtain and my bathroom is not that fancy.

Don't know how effective this will be, but Home Depot sells a Mold preventative product (NOT BLEACH based) that MIGHT help reduce recurrences of mold. I use it from time to time on the boot in my FL washer. The boot NEVER gets moldy, after several years of use. "Moldcontrol" is the product name.

Fast forward to 2019, most hookless shower curtains offer snap-on vinyl or PEVA liners. Polyethylene vinyl acetate (PEVA) is a non-chlorinated vinyl and has become the common substitute in several products on the market. These liners resist mold and easily snap off when it’s time to wash or replace. Both vinyl and PEVA replacement liners are available for sale wherever hookless shower curtains are sold. Further, these liners stand up fairly well to shower spray cleaning products that contain bleach.