Who is responsible for the damages if someone without a license drives your insured car and wrecks it?

You are both responsible. You are responsible because you are the owner and you should make sure that anyone who drivers your car is legal. Also the person who drove without a license is responsible because they didnt have a license which is illegal, and they dont have insurance, which is illegal. You may be the person they go after and you may find yourself having to sue the other person.

Answer

Your policy will probably not cover the event. Check in with your insurance agent.