My kids got them when they started traveling far enough away that I could not easily run over and get them if they got stranded, had car trouble... both about age 16. The card was in my name with them as a user, and I had the credit limit set very low so if they ran it up I could afford to pay if off if need be. The one kid never had any problem with it. The other ran it up the first couple of months, and after that learned to control the spending. They both had to pay their bills. When they turned 18 they could then get their own card. A shared card made it easier also for them to do my shopping such as pick up groceries on the way home from school.
Credit cards if used responsibly are actually a good thing. You get extra protection if you are dissatisfied with a product, and you have a record of your purchases. They make returns easier, because many stores can look up your receipt if you paid with credit card. I have had the credit card company step in for me more than once when a product was not as expected and I had difficulty getting a refund from the seller. Also if you are traveling and your new stuff gets stolen, often the credit card will reimburse.
These days your credit history also helps determine your rates on student loans. Helps to build a history early for those heading off to college.