We took this cruise last month. And I'll have to say it all went very smoothly. The ship was docked at Terminal 88 (although that may vary), and there's police and others directing traffic right up to the drop-off for busses and taxis (we were in the latter). We handed our bags off, putting a DCL tag on each that was provided to us. There's a number of large elevators on the outside of the terminal, but we found walking in and taking the escalators was quicker.
Once inside you'll go through TSA screening, then walk down to check-in at the DCL desks. There you'll fill out a form declaring any recent illnesses, and the CMs will tell you when you can board (we were able to go right on). They gave us our key cards, and a calendar of events for the day. You'll go through one more screening, where your key card is matched up to your passport, and then around the corner for a keepsake photo. After that you board and they announce you! All in all, a fairly painless process.