The answer is up until college graduation. After that, children should be able to afford auto insurance on their own. If they can't, that's the tough reality that they may find themselves in--and the tough reality that parents might find difficult to push their child into. But think about it--for nearly 10 years, you've paid for your child's car insurance. Then what? How long will it take them to transition into paying for it themselves?
After graduation, hopefully your child will receive a full-time position with benefits, and with that full-time job should come a bit of cash flow that will allow your child to afford their own auto insurance. If this isn't the case, perhaps public transportation can substitute for a car.
Some families may make arrangements with their children, such as once they can provide for themselves, then they'll pay for costs like auto insurance. Other families may make an agreement that a child pays for some of their auto insurance throughout college in order to prepare for post-graduation. For others, especially those with larger families, paying for everyone's auto insurance is difficult on the parents.
For children who might be finally out on their own and weighing their options, Arnouts Insurance Agency provides free auto insurance quotes, as well as other insurance information.