It isn't a question of how old your child is, but how mature your child is. You can have some very irresponsible eleven year olds, but at the same time conscientious seven year olds.
As was the case when I was born, many parents have pets beforehand, in which case a child will start learning how to look after an animal from a much earlier age. I was six when my parents bought me pets that were 'officially' mine. I had two guinea pigs, which I helped feed and look after, although my mum did do a majority of the work when it came to cleaning out.
I think if a child is mature enough, then as long as they are they are physically capable of looking after a pet, then it doesn't matter what age they are. If, however, you mean to give the child total responsibility, without the parent helping out much, then I would say ten or eleven.