They don't have covers on them? If not, then I would definitely look into something that will keep the bugs out.

as long as you have screens over the openings (tight mesh like window screen), and the water never fills up near the mesh (permitting access to the water through the mesh) it should be fine. if no mosquito can fly in, it can't breed. if a moquito does manage to lay eggs in the water, the resulting offspring shouldn't be able to fly out.

Mosquitoes can go down the gutters and down spout. They can lay eggs in a bottle cap of water.
I use the dunks in my pond. Fish can not eat the eggs and mosquitoes fast enough. I use them in my rain barrels also. But in a pinch I have put vegetable oil on a rag and suspended it in the rain barrels. It smothers the eggs and mosquitoes.
I also use the BackYard spray.
We are in an area that has lots of mosquitoes. A lot of lakes and ponds and swimming pools. Horse troughs, goats etc. That have to have water for drinking.

Tell me more about the veggie oil on the rag and how and why it works. I have two rain barrels, one is graywater and alas both have overflowed enough to have gone through the mesh & yes mosquitos have increased because of it. I want to get right with my graywater barrels and not breek skeeters!

Thanks for all the tips you can provide to this rain barrel and graywater newbie!