Mosquitoes need 7 days of standing water to complete their life cycle, so yes, this can be a problem if not managed properly. There are several ways of dealing with mosquitoes – chemical means (“dunks” or something similar), biological means with goldfish (make sure the barrel is not in full sun), or physical means with vegetable/seed oil (suffocates larvae) or mild dish soap (breaks the surface tension so adults drown before they lay eggs). Make sure you repeat this every time there is a storm when your barrel overflows.