State laws vary however in general, yes as the car owner you can be held liable for damage a driver causes in your uninsured vehicle. As the owner you have vicarious liability for those that operate your vehicle. If the car is not insured and your state requires auto insurance, or another type of financial responsibility to be on the vehicle, then you should not drive it or allow another person to operate the vehicle and thus knowingly breaking the law.
In an accident normally insurance follows the car first and driver second, so you as the car owner are primarily responsible. If the driver causes an accident then both the driver and you as the car owner can be held liable for the damages the driver caused while operating your vehicle.
If the driver is cited for driving without insurance at the scene of the accident then in many states they can have this ticket dismissed if they give the court the car owner's information for the ticket to be transferred to the owner. Penalties for allowing an uninsured vehicle on the roadway vary from state to state but can include fines, suspension of driver's license, registration, car impoundment, etc.
Also in some states, such as Ohio, you can be ticketed for wrongful entrustment if you allow someone to drive your vehicle without insurance. If your vehicle is uninsured and you put it on the roadway, either by driving it yourself or allowing another person to do so, then you are putting yourself at risk for penalties by the state and financially if the car is in an accident.
Get low cost car insurance quotes here with us if you have an uninsured vehicle that needs coverage.