The cruise terminal parking lots are fenced in, but I can't say that are monitored all week. I assume that are. Cost is like $10 a day. I would recommend EZCruise Parking. They are less expensive and you will need to be shuttled back to the ship from the parking lots either way you go. Google them and make a reservation.
Then you drop off your luggage and/or family at the terminal and have them get in line and then you go and drop the car off and get suttled back...
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise, Magic 1/14, Freedom x 2 6/14
NCL Dream October 2006, RCCL Mariner OTS June 2007, Celebrity Silhouette 2013