Depending on the child, I'd say anywhere from 17-21. If they moved out on their own, then they should be able to do for themselves or learn to figure things out. I'm not saying they cannot ask for help, but more, by that age, they shouldn't expect their parents to drop everything to dig them out of whatever.
I was pretty self sufficient by13 or 14. I set my own appointments and mostly got myself to them and had no problem getting any of my Rxs filled (doctor, dentist, orthodontist, hair salon, mall, grocery store). I shopped on my own for clothes, shoes, school supplies, and occasionally popped into the grocery store. I did my own laundry most of the time and regularly cooked dinner. By the time I moved from home at 19/20,I relied on no one but myself.
My mother never manage to cut my brother's apron strings. Even at 16/17 he wouldn't go somewhere without a family member, not even the mall. Which is why she has her grown bacon-fed knave adult son still living at home.
I have trouble wrapping my head around how we grew up in the same environment.