How do Alaskans handle the long dark winters? Easy--they turn on the lights, of course! Seriously, though...
Have you ever sat in a hot tub with snow all around? Well, the hot springs in Alaska are a large-scale version of this famous Colorado pastime. Combined with a Northern Lights show overhead--what could be better?!
Many people take a hike in their snow shoes, go cross country skiing or sledding on lighted hills and trails. Millions of lakes provide ice fishing & ice skating as options as well.
For those who are adverse to the cold, making a bonfire, cooking s'mores and drinking hot toddies with friends is a great time. Or, just as anywhere else, spending time indoors watching movies or playing games and simply enjoying one another's company is always nice.
Many Alaskans utilize much of their winters for working on their handicrafts to sell during the spring and summer at outdoor markets, fairs and such.
With so much darkness, it's wonderful to turn off all the lights in your house, open the curtains and stare at the aurora! Fairbanks is the ideal location for this. What a spectacular Midnight flick!
In essence, Alaskans behave just like the rest of the world does during the evenings when/where it's cold and dark out. They go out to stores and restaurants and movie theatres. It's also an opportunity to pick up that book that you've been too busy to read, take up a new hobby or scratch-cook some of your favorite winter comfort foods.
Overall, by the time you exhaust all the options above, Spring is peaking at you from just beyond the horizon!
As far as the question that Alaskan's often hear from visitors, "How do you possibly sleep with all of this daylight!??!" -- the simple response of, "Have you ever taken a nap on a summer afternoon?" usually puts things into perspective.