Fairbanks is probably the best place in North America because it's under what's called the auroral oval and has relatively dry, clear skies. Late September through early October is a pretty good time to go, it has dark (enough) nights, milder temperatures than winter, and beautiful fall colors.
That said, if your goal is maximizing your chances of seeing the lights, I'd also consider northern Finland, Norway, or Iceland. Finland (especially around Saariselka or Inari) is one of my favorite recommendations because you can pair the aurora with glass igloos, saunas, reindeer farms, and a really amazing overall experience.
If you choose Alaska, I'd would personally recommend booking a guided aurora tour or staying at one of the aurora lodges rather than driving yourselves every night. The guides know where the skies are clearest, monitor the forecasts, and save you the hassle of navigating unfamiliar roads in the dark.
One last tip: no matter where you go, I'd plan at least 34 nights dedicated to aurora viewing. The Northern Lights are never guaranteed, so giving yourself more opportunity to see them makes a big difference.
Feel free to reach out if you have any questions or if you decide you'd like some help putting the trip together!
Luxury & Adventure Family Travel Agent
Personal Travel Concierge
Europe • Africa • S. America
www.voce-travel.com
Kyle.regitz@fora.travel