I used to live there and I don't know of a place to leave a car. Also, I personally would not be comfortable doing so. Miami is a big city, with big city problems such as crime. Unless I stayed in a hotel the night before and they let me leave the car in the hotel lot, I wouldn't even consider it. Just my 2 cents.
put in MIA or Port of miami and the dates then search.
It will come up with a few place for the airport but only one for the port: CBM Parking. You will save more if you prepay. They have a shuttle service to and from the port. It runs on the hour and when you get back you will have to call in and give your ticket number. This is how you are picked up.
Last cruise there was another couple next to us waiting for the van and when it showed up the driver checked the ticket numbers and then took us as that was the numbers he had the other couple had to wait for the next van.
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First of all, you are lucky to not have to fly down. I think the port parking is expensive too, but not nearly expensive as a rental car would be so take that into account. I have been looking for parking myself, but it seems the previous post was right. CBM is the best option with $12/day for self park or $13/day for covered parking. I feel these rates would be the most affordable you would likely find. Though you may also wish to consider taking a train down the day before. Take into account a couple of fill ups, and parking and see if the cost would be less or more than an airline ticket and shuttle or a train, cab, and hotel.