For tires on a trip to Alaska, if your current tires would be good enough to drive cross county across the Lower 48 or east to west across Canada, then they should be good for a trip to Alaska. Trailers tend to be harder on tires than vehicles. I find one spare tire for the tow vehicle and a mounted spare for the trailer, plus an unmournted trailer tire works for me. I much prefer radial tires all around and stay away from bias ply tires such as many ST tires tend to be, especially as OEM supplied on trailers.
I also carry a tire plug kit and a 12 volt air compressor, a Viair 400, with me on most trips anywhere we go. The only recent tire problems, 2009 trip, I have had was a rock punctured tire on the Top of the World road out of Dawson Town headed for the Alaska border with Yukon. Of course I had not taken my Air compressor on this trip. Also check the air pressure in your spare before leaving home, as I had not done. LOL
Prior tire problems would be a screw in a tire, noticed while getting fuel in Watson Lake and that station patched the puncture on the Class C we were driving.
Tires, like windshields, have been a bigger problem while we were living and now vacationing in Colorado than the years we lived in Alaska and our trips back and forth.