When I used one I drilled holes in top and bottom of the underbed storage box. I didn't cover the cups inside.

Karen

I drilled hole in bottom of container and in the cups i used. then covered with pastic sheeting with a few hole poked into it. but my bins were collected with missing lids.

I ws with lids and without lids using a plastic bag with holes poked in it. Both work well. Holes are needed in the top to let in rain/snow/air and in the bottom to drain.
In your area saran wrap should be okay. I like a heavier plastic for the snow that accumulates on tops here. For those wo lids I slip the s.box into a large plastic bag (those large bags from tp at Costco) with holes in the tops and the bottom of the bag and use clothes pins to keep it closed or put excess bag under neath and weight of box keeps it closed.

This works great with extra 4" plastic pots, seed trays, cups and other containers.

I was going to use a polyethelene cover for the bins with the cover. I was going to make air hole in it. Is that okay?