I would decide when upon arriving in Whitehorse of which route to take :
1 - If weather forecast is great or so so for the next few days or following week, go for Dawson City and Top of the World Hwy route first.....Taylor Hwy on the Alaska side during rainy conditions might not be an enjoyable drive between Jack Wade Jct. and Chicken at mile 66 thru 96.
2 - If weather forecast is bad, just stay on Alaska Highway enroute to Alaska and hope to time it right for decent enough weather upon leaving Alaska via Taylor and Top of World Highways.
3 - (if Top of the World Highway was not first traveled) Upon arriving in Tok enroute to departing from Alaska, check on any potential SEVERE weather and road conditions before heading on Taylor Hwy. and Top of World Hwy. to Canada and Dawson City..... if the Taylor-Top of the World Hwys. have no bad reports, do take on the route.
4 - Since this is the far north we are commenting here, roadways can be subject to change due to potential roadway washouts, wildfire closures, or any other kind of potential road closure that could occur, so do keep up to date with the route when arriving in Whitehorse along the way or arriving in Tok upon departing Alaska if Top of the World Highway was not first traveled.