It's not a matter of age. It's a matter of strength, size and if you're even capable of navigating your neighborhood (or where ever you walk the dog). Dogs are strong, believe me. We always hike in the woods and we met this couple (sporty and full grown) who had three greyhounds who had gotten excited by an animal and ripped the leashes out of their hands and started running howling through the woods. If you aren't stronger than the dog, you're in trouble. Than there's always the chance that the dog decides it really doesn't like you, if you're alone, you're in trouble again. Dogs are predators no matter ho fluffy and cute they may seem. If the 10 year-old just wants to hold the leash with someone to help beside them, that's fine. But I can't stress enough how dangerous animals can be when they want to.
As old as it takes to get someone to let you walk them. Read up about the dogs you may be walking and make sure you have the strength to control them. Also have a price you will charge in mind. Get some price info from others and be prepared to clean up after they potty.