Parroting some of what romore suggested, you could try the Cassiar Highway (37) up through central BC to Meziadin Junction, then 37A west to Stewart BC/Hyder AK. As for YK/NT toe dipping, Fort Liard NT would be the closest point in the Northwest Territories, and then west towards the Yukon, but you might have to back track to Fort Nelson and then northwest towards Watson Lake and Whitehorse. Or you could go north from Jasper to Fort Liard NT, then do the back track to Fort Nelson and head north and west to Whitehorse and Haines Junction, then take the Haines Highway south to Haines AK.
I wouldn't think any of these routes are a "day trip", though. The highways in northern BC and the Yukon last year were usually 2 lanes wide/one in each direction, posted at 90kph mostly, with slower "pilot car" guided one way travel through regular construction zones, that could be miles in length on gravel or worse, and not many amenities in some stretches. There are no interstate type highways up there. Be forewarned.