From Cambridge Dictionary:
Don’t you "care about" what happens to the children?
To paraphrase your example, you could replace the "what happens to the children" with a noun phrase (e,g, Don't you care about the children/the children's future/well-being, etc.). There must be an overt or underlying object (usually a noun phrase) following the preposition.
How about "I don't care what happens to the children?" Now "What happens to the children" becomes a complement to the verb "care" and… the meaning changes.
"I don't care about…" entails lack of consideration or importance, whereas "I don't care" entails immateriality of choice or no preference or interest based on liking or partiality for something (for what happens to the children, for example).
I don't care what Biden will do. (I don't care what he will or won't do. He belongs behind bars.)
I don't care about what Biden will do. (???)
So, it's not the informal style that lets you drop the preposition. It's not one of those "both are OK" situations either.