When I did this in December 2008, there was a shuttle bus that was operated by the airport between the car rental facility and Port Everglades.
HOWEVER, not everyone was eligible - the bus driver was asking people if they had rented from certain car rental places and turned some people away. I had rented from National, but honestly don't remember if National was one of the allowed agencies or not. I simply answered yes and got on the shuttle
I'd call National at the airport to confirm the current situation.