I think besides having window screening tightly over each bin, that traps are the way to go. Even with the screening they can get in there, I don't know how but the Safer's traps are the best for me. I got another brand that looked a little different and never caught one moth while the Safer's one filled up.
There's a youtube video about cutting the bottoms out of those drawers but basically you have to use a utility knife and cut gradually. Like score it, then make the score deeper with each cut. It isn't easy but if you try to make a cut all the way through first try you'll shatter the plastic. Ask me how I know that