Answer:
To prevent car or travel sickness, Dramamine is perfectly safe for use in dogs and cats. The normal dosage is 2-4 mg per pound of body weight given 30 minutes before travel every 8 hours.
The following advice given by another reader does pertain to general health - but certainly does NOT apply to the prevention of car sickness or travel sickness. I run a dog shelter and do not subscribe to the below panicky sound of getting your dog to the vet ASAP. Many dogs vomit once or twice and it is no biggie. They even eat grass to cause themselves to vomit to clear an upset stomach. However prolonged copious vomiting of more than a few times, or vomiting with blood, or vomting AND diarrhea should be checked by your vet.
This applies only to general health as explained above - "You should not be giving your dog Dramamine. Is your Veterinarian aware of this? If your dog is vomiting, there is a reason for it. You dog needs to be taken in to see your Veterinarian to be checked out and treated properly. Do not treat the dog yourself as you could cause more harm than good. I highly recommend you get your dog in to see your Vet ASAP. Vomiting is bad and can dehydrate a dog fast then they die."