Whenever school is out, prices and demand go up. So Christmas/New Years is the most expensive time of the year, followed by Thanksgiving and Spring Break, then the 2nd/3rd week in June.
The best prices can be found the 2nd/3rd week in September because everyone is going back to school and back to work. Plus, it's the middle of hurricane season, which tends the scare alot of people away. But this is when we go because we get really good rates.
The 1st/2nd week in December can offer some good rates as well because families want to travel for Thanksgiving and Christmas, so not alot of people plan to travel at that time of year.
With that said, if you want a cruise with more adults and less kids, then use this to help in planning;
1) Choose a more upscale cruise line. For the most part, families want a cheaper cruise with lots of amenities. Of the mass-market cruise lines, Celebrity and Holland America normally draw an older crowd and their costs aren't much higher. Any of the luxury cruise lines (Crystal, Regent, Seabourn, Silversea, etc.) will always have very few children, but costs are higher.
2) Choose a longer cruise. Families like to stick with 7-nights or less.
3) Go to someplace different. For the most part, families like warm weather and nice beaches. Most kids don't necessarily like Alaska, Canada/New England, or Baltics cruises.
4) And the obvious one, don't choose a cruise when school is out.
In your situation, you really can't put #3 & #4 into the equation, so you're left with #1 & #2. Of course, if you can't do a cruise longer than 7-nights, then you're pretty much stuck with the #1 option.
Now, the thing this year New Year's day is on a Tuesday, so most schools will be back in session on Wednesday or Thursday (at least that's the case around our area), meaning there could be more kids onboard New Year sailings than in some years. We've found alot of parents don't mind keeping their kids out of school for an extra couple of days for cruises around New Years. We have more families booked this year than usual because schools in our county start on Thursday, so it only means an extra two days out of school.
But whatever you decide, do it quickly! Prices have already gone up tremendously and availability is becoming more limited. Holidays are when you really want to book as far away as possible.