The answer to your question varies as individuals are different.
Some parents prepare their children to be responsible and independent at an early age, by teaching them lifeskills and increasing personal responsibilty as they mature.
Some parents fail to instill lifeskills, and thus, should help support their children until the child acquires the ability to function successfully on his/her own.
Having said that, I personally believe a child should be able to function in society productively by the age of 18, even if they are in college. If a child is in college, I think it is appropriate for a parent to still support the child. If a child is not doing anything to improve his/her own life after they graduate high school, I think parents inflict more harm than good if they refrain from placing requirements: getting a job, volunteering, community service, etc.
I have a cousin who earned his college degree on a soccer scholarship. When he graduated, the job market was tight, to put it mildly. He moved back in with his parents. He is now 23, and is saving money to move into his own place. He said at a recent family gathering, "It's just time. It's part of growing up and becoming a man. My family is great and all, I have a wonderful situation. But, I just need to support myself and be out on my own." His goal is to have saved enough money to live on his own comfortably for a few months. He will continue to work while looking for employment in his specialty.
I think anywhere from 18 to 24.