Have you been to Alaska or the Yukon before? What do you want to do in Fairbanks and on the Alaska Highway?
I just flew into Fairbanks in June for a conference, but I have been there many times since my first trip in 1972. Fairbanks to Beaver Creek, YT, on AK2/YT1 is, IMHO, not the most scenic part of the Alaska Highway route and is definitely the roughest/slowest from the standpoint of frost heave damage to the road. The AH section around Klune Lake and the Haines Cutoff are much more scenic and generally smoother/faster if you only have a short time for your trip.
If it is too expensive to fly into Whitehorse, look at airfares to Juneau. They are generally about the same as Fairbanks airfares. After touring the Juneau area and perhaps taking a boat tour to Glacier Bay, take the AMHS ferry to Sitka (my favorite town in Alaska), back to Juneau, and then to Skagway (my least favorite town in Alaska).
In Skagway, you can ride the White Pass & Yukon Railroad to Lake Bennett or Carcross and/or hike the Chilkoot Trail. Then you can rent a car and drive to Whitehorse and Dawson City, if those historic Klondike Gold Rush locations are of interest to you.
This combined ferry/car tour gets you a brief introduction to both SE Alaska and the Yukon. But you miss Denali NP, Valdez, the Kenai, Anchorage, and other prime tourist destinations in Alaska.