We've taken several cruises out of NYC, including Carnival Victory this past summer.
Your ship is scheduled to depart from the Manhattan piers. A great site to check out is http://www.nycruiseterminal.com/terminal.html
Is the organizer of the singles cruise providing transportation to/from the hotel? And, have they told you the hotel and its location? That would influence the choice of airport for me.
We tend to use LaGuardia airport (LGA). Taxi fares from LGA to midtown Manhattan are generally in the range of $30-$35 (depending upon traffic) plus another $10 or so for tolls and tip. Kennedy airport (JFK) is another option, but taxi fares to Manhattan are higher -- a flat rate of $45 PLUS tolls and tip. I have not used Newark airport (EWR), but hopefully someone will come along who has. But, generally, I would select LaGuardia.
I am originally from NYC, and I now travel there quite frequently for business, so I'm happy to answer what I can.