There are New Year's Eve packages on that page.
If your time is short, Seneca Lake is probably where you should focus.
Wineries that are my personal favorites, in the whole region, are...
Dr. Konstantine Franks
Bellwether Hard Cider
Earle Estates Meadery
Winter, is a pretty quiet time around the Finger Lakes wineries - many might only be open during the weekends. Having said that, there is a lot to be said for visiting when things are calm and you can enjoy the wine, chat with the staff, etc., which is really hard to do on summer weekends. Do some seaching to find the places you might be interested in (there are some threads here) and give them a call regarding New Year's opening