School district near Fort Drum are Indian River(2/3rd's to 3/4ths of the base kids go there), Carthage(the other quarter or third of the base kids go there), Watertown(which has a good bulk of the military owned off post housing), Lowville, 1000 Islands, Gouverneur and Copenhagen. There is military owned off post housing in places like Calcium(IR schools), Evans Mills(IR schools), Philadelphia(IR schools), Carthage(Car. schools), West Carthage(Car. schools), Clayton(1000 Islans schools), Gouverneur, Lowville, Copenhagen and Watertown. This might have changed since I was stationed there and it might be privately owned housing now.
There are other school districts nearby that you could find housing in too. General Brown in Dexter is probably the "best" school district along with 1000 Islands in the area. Kids from the villages of Glen Park and Brownville, along with a good portion of the town of Pamelia go to school at GB too. Copenhagen is another very good area school district. You can get a good education at any of the school districts in the area(Jefferson and Lewis Counties, along with Gouverneur in St. Lawrence County).
As for things to do, during the summer, you can go to places like the 1000 Islands, which have tours and festivals, along with the village of Alexandria Bay. Sackets harbor is another village with a nice little downtown and a historical site. There are a couple of beaches a long Route 3 just west of Sackets Harbor that are state parks. Keep in mind there has been a proposal of closing some state parks, but there isn't anything offical as of yet. There is a mall in Salmon Run Mall and much of Arsenal Street has been built up since I ETS'ed in 2003 from Drum.
It is a safe area and while there is crime, there aren't any "war zone" areas in the city of Watertown.
Like jblake said, Syracuse is only 70 miles away and is definitely doable on weekends, even during the winter, depending on the weather in between. Some soldiers even live in the Syracuse area and take a shuttle form the Air NG post to Drum(or at least they did for a while).
If you have a passport, you can check out Canadian cities like Kingston or Ottawa, their nation's capital. Gananoque is a nice resort town on the Canadian side of the 1000 Islands too.
Here's some good information that will help you decide:
New York schools - NY elementary, middle and high school information
New York schools - NY elementary, middle and high school information
Gouverneur Central School District Overview | GreatSchools
Immaculate Heart Central Schools
: Home
CAUGUSTINIAN.ORG
St James School
Fort Drum, NY Housing Relocation Information
Fort Drum Mountain Community Homes - An Actus Lend Lease Community
watertown apts/housing for rent classifieds - craigslist
Fort Drum | Welcome
Fort Drum Housing, Fort Drum Family Housing, Afforadable Apartments in Fort Drum - Luk Properties
http://www.drum.army.mil/sites/local/
Military.com - Finance - 2010's Best Cities for Military Families (http://www.military.com/Finance/content/0,15356,207960,00.html - broken link)
Fort Drum Regional Liaison Organization
Housing Fort Drum
Fort Drum Apartments for Rent - Fort Drum, NY on ApartmentGuide.com
Watertown, NY - Official Website
Greater Watertown North Country Chamber of Commerce, Watertown Chamber of Commerce, Watertown New York
Thousand Islands | 1000 islands | Tourism Info | Images Thousand Islands Region
VISIT THE 1000 ISLANDS
Uncle Sam Boat Tours: 1000 Islands Boat Tours in Alexandria Bay NY
Official Boldt Castle Web-Site - Home Page
Thousand Islands Life Magazine Current Issue
Welcome to Your 1000 Islands NY Guide - Alexandria Bay Chamber of Commerce
Alexandria Bay, NY
The Village of Sackets Harbor NY - Experience the Treasures!
Thousand Islands Region - NYS Parks, Recreation & Historic Preservation
Watertown, NY Jobs | Jobs in Watertown New York and Surrounding Areas
Welcome to Samaritan Health System
https://stream.ats.hrsmart.com/cgi-bin/a/searchjobs.cgi (broken link)
http://www.nyab.com/job/default.asp (broken link)