I plan to look at this more fully at another time (I'm currently traveling and unable to write a detailed analysis), but for now I'll put up a couple of charts and invite comment. First, the fraction of the hourly sky condition observations that reported "clear" in deep winter (Dec-Feb) since 1950, see below.
One issue to be aware of here is that the automated (ASOS) observations were introduced in 1997, and it's inevitable that this change produced a bias relative to earlier years, because the ASOS ceilometer doesn't detect clouds above a certain height (12000 feet in the original ASOS ceilometer). For example, here's a comparison of ASOS and conventional observations for several stations in the lower 48 over a limited period of time (taken from a report here).
These results indicate, as we would expect, that the ASOS instrument over-reports clear skies compared to earlier manual observations, so if anything we would expect to see a higher frequency of clear skies since 1997 in the chart above. It seems that the change in observing procedure does not explain the apparent decrease in frequency of clear skies in recent winters in Fairbanks.
Another potentially confounding factor is that fog in winter in Fairbanks obscures the sky and precludes cloud cover observations, and we know that fog used to be more common in Fairbanks (see e.g. here). However, we would expect that fog forms preferentially under clear skies in winter, so presumably the missing observations from foggy times in earlier years would be more of the "clear sky" variety. Again, the trend towards less fog (more complete cloud cover data) doesn't seem to explain the decrease in frequency of clear skies.
Another way to look at the data is to calculate the mean cloud cover fraction from the reported coverage category (i.e. clear, scattered, broken, or overcast); see below. I hope to post more discussion and analysis at a later date.
Charts added by Brian on 11/2 & 11/3