A 30-06 is generally considered good, or maybe optimum for northern game, in general. There is no field difference between a 308 and a 30-06, so fly at er.
I have seen most of the Yukon game, including grizzly, never hunted there, but have hunted and taken game just across the border in northern BC.
I one time went on a day prospecting trip north of Carmacks. I was looking for a mineral outcropping I had instructions to.
I started out with my day equipment, but no rifle. I shortly started to see grizzly tracks. Then more, and fresh, grizzly tracks, so back to the vehicle for my 30-06. I saw more grizzly sign that day than I have ever seen in a day in BC, but none of them were exceptionally large.
It was fall and I was a bit concerned that I could be accused of hunting, without a licence and a non resident, for good measure.
Later I met a RCMP officer and I told him about my concern. His answer, at that time, was don't worry. You should have a rifle in that country!