I found a lot with decent reviews by going through ParkwayParking.com
There is a shuttle to the port that runs frequently and I am paying a bit less than $10 per day. We don't sail until 24 December, so I can't offer my own review yet.
Update: The parking lot we used was Safe Cruise Parking. True to form for a lot of these operations, it is in a sketchy part of town. But we were a 5 minute drive away from the port. The shuttle took us on a short tour of the port since we were the last stop. We still made it to our ship in about 10 minutes. The parking area was walled in and seemed "reasonably" secure, but it was neither Fort Knox nor patrolled by armed guards. Pick up at the port was easy. As we left the ship we were greeted by drivers iffering rides, I showed them the bright yellow sticker I had received when we parked and they directed me to stop #9. A short wait later the van showed up an we were on our way back to the car. Aside from dozens of birds mistaking our car for a statue (we headed straight for a car wash) there were no problems.