In the midst of the Cold War, I could see Greenland becoming a pretty heavily militarized outpost of the US. Even OTL, Thule AFB was an important strategic bomber base. Despite the cold, one could see it being used as both a missile base and sub base as well.
It could also play an interesting role in the development of transatlantic commercial aviation as a hub for US-Europe traffic if the old Bluie West WW2-era airbase at Narsarsuaq could be made suitable for commercial flights (located in a fjord in the southern part of Greenland, it's a notoriously difficult approach). You might see Pan Am and/or TWA use it as a transit point that would allow them to overfly Gander each way and stop there instead. While it's a bit off the Great Circle route, it would be within range of the long haul prop airliners of the day to most of the major cities on the East Coast and Europe. You could see a good chunk of the US East Coast having one-stop service to Europe a decade or so earlier than OTL and a much greater US presence at more European airports.
By now, it would probably largely be in disuse, but it could still be an important military resource as an airbase for deploying troops overseas and as a US territory or state, would no doubt still be getting air service between the US and Europe.