It will vary depending upon state laws but in general both you and your cousin could be held responsible.
You could be held responsible for knowingly allowing someone to drive your car which is without insurance or plates. And your cousin could be held responsible for driving a vehicle that he knew was unregistered (without tags) and was not properly insured.
In some states the tickets for these infractions would be written to him directly, while in other states the tickets would be written to the car, thus you the owner. And in other areas of the US the officer can decide which gets the tickets.
If the tickets were written to you cousin in many states he could give the court your information as the owner and pass on the ticket, in part or in whole, to you since you allowed him to take your car on the roadway illegally. The penalties for driving without insurance or plates vary by state but are usually quite harsh and can include things such as driver suspension, impoundment of the vehicle, jail time and fines.
Also keep in mind that if your cousin was in an at-fault accident in your vehicle then both he and you would also likely be held responsible for the damages he caused. If a person in another vehicle sustained bodily injury then in most states you both would be held liable and have to pay back the damages to the car and for the person's medical expenses, which tend to become costly very quickly.
Due to all the problems that could arise from letting your cousin drive your old car that is without insurance or tags it would be in your best interest to not let him drive it. If you no longer need or want the car you may want to consider signing the title over to your cousin so that he would be the owner and responsible to get the car registered and insured. If you did this though make sure he goes to the DMV and puts the car officially in his name so that you are no longer named as the owner of the car.
If you or your cousin would like a quote to see how much insurance would cost for this vehicle, click here for an affordable auto insurance quote.