A cruise ship is a perfect place to celebrate New Year's Eve. I would recomend the Solstice class Celebrity ships.
You will find children on any cruise line on any ship during the Holidays, but I did do a New Year's Eve cruise and it was a blast.
Carnival Liberty New Year's Eve 2007
Liberty of the Seas 5/2008+11/2009
Allure 1/16/ 2011
Independence of the Seas 12/29/2013 Top-notch!