for the daytime aboard the ship, cordoroy, jeans or fleece pants, a turtleneck, fleece, sweater or sweatshirt layer. You will need some sort of winter parka, hat and gloves.
For evenings - New Year's Eve will most likely be a formal night. A nice cocktail dress or dressy suit will do you just fine.
For the other nights - wear what you would wear to go out to dinner at a nice restaurant back home.
What are your other ports of call? You will need to pack summer clothing if you are heading to the Bahamas or the Caribbean.