Wonderful question. I'm not aware of any international diplomatic difficulties around 1959. Subsequently, however, Alaska statehood would prove to give its residents powerful leverage in the debate over America's offshore fisheries. In the 1960s and 1970s, foreign fishing fleets would come right up to the 12-mile line internationally accepted as territorial waters, which caused a great deal of heartache in Alaska.

As a result, the United States created the Magnuson-Stevens Fishery Conservation and Management Act in 1976. This extended the U.S. "exclusive economic zone" to 200 nautical miles offshore, a critically important distance for Alaska, which exports more seafood than the rest of the United States combined. The 1982 United Nations Convention on the Law of the Sea formally adopted this 200-mile limit internationally, and subsequent agreements between the United States and Soviet Union settled fishing boundaries along the international date line.

What's more interesting in the 1959 timeframe is the Cold War's effects on Alaska statehood in the mind of President Dwight Eisenhower.

Eisenhower was (and remained until his death) an opponent of Alaska statehood. In the latter 1950s, Eisenhower doubted Alaska's economic viability as a state (oil not being discovered in economic quantities until 1957 on the Kenai Peninsula) and believed that it was crucial to military defense.

Eisenhower at first favored partitioning Alaska; leaving the sparsely settled western and northern portions as a territory, a de facto military reservation, in order to provide what he thought was needed for defense. This was opposed by Alaskans. As a compromise, it was agreed that Eisenhower would have the ability to withdraw a vast tract of land, as he wished, from those areas in order to serve military defense. That would have the advantage of preventing the Soviet Union from knowing where the military installations would be in advance, and it would meet the demands of Alaskans.

Terrence Cole's Fighting for the 49th Star is definitely worth reading; it's much better than Whitehead's Completing the Union.