i wasn't rich as a kid. i always earned my own money, and while i may have only had a couple hundred in the bank, i still had checks.
it makes no sense for your bank to place legal or age restrictions on your ability to get a debit card. you cannot over-charge on a debit card like a credit card, and it's safer than carrying cash around. moreover, it's your money and you have the right to access it any way you choose. shit, i know pre-teens with debit cards.