Yep Dramamine is safe - just follow the dose directions your vet said. There's two different possible reasons for it - anxiety or actual motion sickness. My Beagle has motion sickness, no anxiety what so ever - she actually loves car rides

but she'll drool badly then vomit ever since she was a puppy even if no food for a while prior, head out the window, nothing worked with giving her her something. I haven't tried dramamine yet, but I was also told by a vet that it's safe. If your bothered by the possible drowsyness though, something I've had perfect success with is an herbal remedy called
Coculus - it usuall comes as these tiny round pellet type pills. There are a few homeopathic online retailers that sell it but I found it at my local health food store

It works like a charm and no drowsiness or worry about adverse side effects plus it's natural unlike Dramamine