The requirements for bringing a dog into Mexico are quite specific. If you are arriving by car, you are allowed to bring two dogs (or two cats) into Mexico, per car. You also need to have an International Veterinary Certificate, which is only valid for 30 days and costs about $70. Those are the rules. However, they are rarely enforced.
I own an ocean-front condo in Puerto Penasco (also known as Rocky Point) and we drive down there every six weeks or so. And yes, it is perfectly safe. It is an easy, four-hour drive. We cross the border at Lukeville, a sleepy little Arizona town that probably wouldn't exist if it wasn't at the border. Unlike places like Laredo and Nogales, people don't get shot at Lukeville. There's a parking lot, a gas station, a post office, a restaurant and a shop. That's Lukeville.
When you cross the border -- after the border agents on the American side ascertain that you're not carrying weapons or more than $10,000 -- you come to a sign that flashes green or red. If it's green, you just keep going. Welcome to Mexico! If it's red (which hardly ever happens to me) you have to pull over beneath a metal canopy and they ask you where you're going and sometimes, they want you to pop your trunk. They are unfailingly gracious and polite. Then, Welcome to Mexico! They really do make it easy, especially at Lukeville.
So what about your dog? In the seven years I have been going to Rocky Point, nobody on the Mexican side of the border has ever asked to see my International Veterinary Certificate, which is just as well because although I used to have one, I no longer bother to renew it. If you're green-lighted, it doesn't matter. I have been red-lighted a few times, but on those occasions the agents have always been more interested in what I had in the trunk than in the dog sleeping on the front seat.
If you're flying in and going through Customs at a Mexican airport, you might need the International Veterinary Certificate. (I'll be doing that in January, and I'll let you know) But I think if you're driving, a copy of your dog's vaccination record is probably enough. And you do need to have proof that your dog's rabies shots are up to date when you come back into the United States -- although nobody has ever actually asked me for that document, either. However, it would be very irresponsible to take a dog whose rabies shots aren't up to date to Mexico, as there are lots of stray dogs running loose.
If you do hit a snag, be courteous and contrite. And smile a lot. In my experience, most Mexicans will bend over backwards to help you, if you're nice. And Mexico is really beautiful.