My husband spent time learning from a man known in our area as "the worm man" and "worm wrangler." This man is all about studying the most effective and natural methods for vermicomposting.
This worm farmer had done several studies with various lids/covers for worm bins and he found that carpet was the best option to place on top of the compost pile. Overall the best option was using 2-3 slabs of carpet or weighing it down with a large rock or something heavy. He found that this method was best for helping worms flourish and multiple faster. He was realizing that worms want what is most natural which would be soil/sod, so the closest match is carpet. That way it's dark but also has some air flow. You want the yarn (actual carpet) side facing up.
For our outdoor bins we cut a slab of carpet to fit the tops, then lay down a small slab of flagstone on top and it works great! When we had a bin indoors we used a large cooler and had a small slab of carpet on the top with the lid open. You really shouldn't have a problem with the worms crawling out but if you do, use the cooler lid instead.
If you're using plastic tubs then you can use the lids that go along with those.
If you're using a cooler than you're better off keeping the lid off. Unless you're up for drilling small holes in the cooler then you don't want to be sealing off the cooler completely, you need air flow!! Use a carpet slab on top of the cooler if possible or a piece of cardboard!
To learn more about vermicomposting take a look at the post A Guide to Vermicomposting.