The name on the room/cabin has to match the name on the credit card. If you are all sharing a cabin, then your card should be fine. If your kids have a cabin in one of their names, I will say probably not. When my husband and I cruise, I put my credit card on file, and we have never had a problem. I used the same card for my mom, in a cabin with my MIL, and it was a problem.
I put my card on my mom's account in her room, along with a couple of gift cards. On the last day, she purchased a few photos to finish using up the credit she still had from the gift cards, but in doing so, put her about $20 in the negative. Since it was the last day, the guest service desk was crazy. I went to the kiosk and tried to pay what she owed so we could leave but it would not accept it. They had phones around the kiosk area that would connect to Carnival's customer service. I called and that is when I found out the name on the card had to match the name on the room. I ended up buying a gift card and applying it towards her balance.
When I set up her account online before the cruise, when I put in my credit card information that included my name, the website accepted it so I had no way of knowing it wouldn't work until it would not work.