Me and a friend are going to be driving from Anchorage, Alaska to Seattle, Washington in mid-September. Will we be encountering snow at that time of year, or should we be safe?
Picking Dawson City as a representative worst case, you can see from here and here that it can snow a bit as early as late August, but the average snowfall for the entire month of September is less than 2 inches. From experience not too far south of there I can tell you it is generally warm enough during the day in September that any snow is unlikely to remain on the ground very long.
A little bit of snow is not very dangerous in any case as long as the vehicle has half-way reasonable tires (I.e. not big, wide slicks). It seems like the probability of a problem is pretty low.
The Alcan (or Alaska Highway as it more properly called these days), can experience snowfall any month of the year. Having driven it several times a year for the past 20+ years, I have seen it closed due to snow at some point in every month but July, so yes snow closures are a possibility in September.
Several areas of potential concern would be Steamboat and Summit Mountains, Muncho Lake, the stretch along the Kluane Mountains from Haines Junction to Lake Creek. The problem occurs when snow during the day melts then freezes at night. Fortunately the RCMP will close the road if conditions are bad.
How to be prepared: make sure to leave enough time to reach your final destination in case the road does get closed for a day (which can happen for a variety of other reasons too); bring your chains from Alaska to be safe.
You won't need your winter tires, just a decent set of tires. There will be several stretches where you will be driving on dirt (or mud if it has been raining or snowing).