The cruise lines usually offer transfers from the airport. There are a variety of services that can be offered with the transfers. Some cruise lines will get your luggage for you at the airport so you do not even need to go to Baggage Claim. Others will take your luggage after you have retrieved it from Baggage Claim. You usually have to sign up for this service 3 days to a couple weeks prior to the cruise to pay in advance. You may be able to purchase your transfers at the port, but should verify this with the cruise line in advance.
Car rentals can be a more economical and independent option, giving you the choice of when to arrive at the port.
If you are considering renting a car to go to the port, I would recommend a one way rental rather than paying for the rental and parking while you are on the cruise. You’ll want to check out which rental companies offer drop off with no fees at different locations. Also, you’ll want one that will offer a no-cost ride to the port.
Your other option would include taking a taxi or using a town car/limo service. This decision should be based on which port you are departing from and how many people you have. In Miami, Tampa and Fort Lauderdale, the cost for a taxi is not outrageous but at Port Canaveral when arriving in Orlando, it is quite expensive for a taxi. Many times a town car will be lower cost than the transfers with the cruise line as the town cars charge per trip, not per person. If you have 3-4 people, it is usually cheaper to use a town car.
The other advantage to taking a town car or rental is that you can come in a day or so early and stay near the port. The transfers with the cruise lines usually will just be for the day of the cruise, unless you have a pre or post cruise stay booked through them.