Today I drove the Watson Lake to Whitehorse section of the Alaska Highway. All the time I could not help thinking of your question, regarding the mountain passes, and heights. From memory, I think this section is probably typical of much of the highway. Anyway, this section was less than 300 miles, so not quite 20% of the Alcan.
Guess I had never noticed the things you had asked about, or maybe it is because they did not bother me... but today, I took note, and a few pictures, so you can see for yourself.
Think of this road as a mountain road, rather than a collection of passes. In fact, when the road crossed over the continental divide, the sides were probably less noticeable. But it is a mountain road, and as such, it goes up and down most of the time, much more so than Interstates do.
At all times the road appears to be elevated well above its surroundings. In fact, there is a trench on either side of the road. This varies from something like six to 10 feet in places, and well over 20 feet in others. But it is always there. Rarely, other than at rest areas, are you able to drive off the road. On the other hand, for the most part the shoulders are broad, and easily accommodated my vehicle to take the photos below. Even when I pulled off adjacent to double lines, I was able to get well off the road.
Where the sides of the road drop away steeply is typically where I will avoid pulling onto a shoulder, no matter how wide it is. In places where it is a sheer drop, and when it is extremely high, then barriers are in place. But these are not frequent.
However, this section of the road also provides a bit of entertainment for many, and depending on the ages of your children, they may like to take part in this as well. I have posted one such photos below.
[I recall when my older daughter had a great fear of heights (and many other things), we used to sit her behind the driver, or in the centre back seat... but never on the passenger side of the vehicle. This helped a great deal, as she was not always able to see the edge of the road. Thankfully she grew out of it. lol]
This shows the gradual slope of the bank, even though it may be 6 or more feet deep. I was able to park completely on the gravel, well off the road. (And my vehicle is not small!)
Another view showing the trench along the side of the road. In corners like this, it is usually steeper than along the straight.
Not a very good photo, but the trees in the foreground would be some 15 - 20 feet below the road. So you can see how high the road is. Once again, the bank slopes and I am parked well off the pavement.
Here you can see how, particularly in corners, the bank is quite steep and deep. I estimated this to be around 10 feet.
And here is just one sample of the many I saw today. Names, place names and even whole messages written out in rock. There are plenty of places to park safely beside the road, on the gravel shoulder... and apparently many have done so.
Hope this helps.