You should get your credit report and apply for a credit card according to your credit score. If your child is under 18, you can order an additional credit card to your credit card, so you will be the cardholder, but your child can use the card limit.
If your child is over 18, you can become a cosigner for him or her, so you will apply for a credit card and you both will be responsible for it, but it will be him/her who will have to pay off the balance.
But before you do something, you should call the bank you are going to apply to and ask them if the first or the second variant is possible.
All credit cards terms, fees and rates mentioned in this article/post are actual on the posting date. See the current products’ Terms & Conditions on the issuing banks' websites.