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.
Disclaimer: This editorial content is not provided or commissioned by the credit card issuer(s). Opinions expressed here are the author's alone, not those of the credit card issuer(s), and have not been reviewed, approved or otherwise endorsed by the credit card issuer(s). Reasonable efforts are made to present accurate information, however all information is presented without warranty. Consult a card's issuing bank for the terms & conditions.
All rates and fees, and other terms and conditions of the products mentioned in this article/post are actual as of the last update date but are subject to change. See the current products' Terms & Conditions on the issuing banks' websites.