My dogs wear their harness and I attach that to the seat belt with a clip I bought
We have a dog cage for our dog. It is also safer for the dog than to have her him loose in the car
I think the safest way to travel with your dog is either a cage (don't forget good ventilation & water) or by using a special dog harness. If you're worried about a bit of dirt or a few dog hairs, then don't put your dog in the car.