Just a few weeks ago I crossed over in Eilat and it was quite easy. As said in previous posts, you can take a bus down to Eilat and then hop a short taxi to the Jordan border. Or if you have a rental car already, you can just drive to Eilat and return it to the agency in Eilat since most of the major rental agencies have offices in town.
Once you cross over the Jordan border, there will be plenty of taxis that can take you up to Petra. I paid 45 JD one-way, per car not per person, from the border up to Petra. Once in Petra you don't have any need for a car as everything is walking distance or short taxi rides.
Also, depending on how long your trip to Petra is you can just leave your car at the Jordan border. There is parking on the Israel side so you don't need to drop it off in town if you don't want to.
Good luck and enjoy Petra, absolutely beautiful!