The Hyatt is the most convenient for staying the night before. If you are arriving early, do not put your DCL luggage tags on. You'll be responsible for picking them up at baggage claim and bringing them to whichever hotel you're staying at. If it is the Hyatt, you never have to leave the airport. Rates are usually around $180 per night. When you check-in, let them know that you are with DCL and they'll give you a packet of information. Before you go to bed that night, put your luggage tags on and leave your bags outside your room. A representative will pick them up and you'll next see them sitting outside your stateroom door on the ship. In the morning, just take your overnight bag down to the hotel lobby and meet the DCL representative.
We've also flown in a day early and stayed at the Hawthorne Suites Airport. They're rate was about $99 and they had a complimentary shuttle to and from the airport, plus a complimentary hot buffet breakfast in the morning.
There are so many options when you decide to fly in the night before. Many guests do a one-way car rental and head right for the port. There are many hotels in that area and you could spend some time on the beach. We travelled with a family of 5, so it actually worked out less expensive for us to rent a car and skip the transfer. That way, we could also stop at Walmart and stock up on some water and beach toys that we didn't want to haul down in our luggage. A limo or towncar is also a nice splurge to start your vacation.