Mushroom flies in particular can be a pest, but usually only occur if you have lots of grow bags in one space. Slugs also like to eat mushrooms but if you place them up away from the floor this will prevent them getting in. Most other insects are not really bothered by mushrooms.
You can place some fly screen material over the holes if you want to exclude all insects, or install a couple of stickly fly strips in the box to catch and kill and flies that come near.
It’s not usually a problem with just a few bags, but if you have more bags over time it could be a good idea to use fly strips or fly screen.
best wishes, Adam