I watched this fantastic documentary called the Wings of Russia video series, and they spoke about how during the early days of the cold war, there were a number of airfields near the north pole, as it allowed for easier access in the event of nuclear war. Both sides did this to some extent, so was there ever any kind of conflict from having such forces in such a barren place?
Yes, and some resulted in fatalities. Generally, however, aircraft operating in the Arctic and along the Alaska-Soviet border were engaged in gamesmanship and testing their opponents' responses without violence. There were important exceptions to this, particularly in the first years of the Cold War.
Between 1949 and 1956, the United States suffered a critical lack of intelligence about Soviet capabilities. The American intelligence service had almost no on-the-ground sources of information, and so the United States and its allies were forced to rely on electronic intelligence gathered by signals interception and overflights by reconnaissance (spy) planes. Until the U-2 entered service in 1956, these overflights were performed by lower-flying, slower-flying aircraft capable of being intercepted and shot down by Soviet forces.
The years of 1954 and 1955 were particularly hazardous. Between September 1954 and June '55, four American aircraft were shot down by Soviet aircraft. Others were attacked without being downed.
On May 8, 1954, the U.S. Air Force flew three RB-47 reconnaissance aircraft (variants of the B-47 strategic bomber) from England to the Barents Sea. Two of the bombers, reserves, were kept back while the third overflew the Kola Peninsula, looking for Soviet bombers on the ground. MiG-17 fighters responded to the overflight and opened fire on the RB-47, which returned fire from its tail gun. The American bomber didn't score any hits, but the Soviet fighters did before breaking off as the American bomber escaped over Finland. To avoid ditching, the American bomber had to get an emergency refueling from a tanker.
A few months later, on the opposite side of the Soviet Union, things got worse. On June 22, 1955, a U.S. Navy P2V-5 Neptune was cruising off the coast of the Soviet Union when it was intercepted by six MiG-15 fighters that raked the plane with cannon fire. The plane skimmed the surface of the ocean to drive them off, but the damaged plane's wing caught fire, and the pilot was forced to ditch on Alaska's St. Lawrence Island. A surviving member of the crew visited Alaska last year.
Once the U-2 began its overflights, the need for these dangerous patrols diminished significantly. When the U-2 shootdown took place, the Discoverer series of spy satellites was just coming into its own and was already replacing the need for U-2 information. After the U-2, the rate of violent incidents in the Arctic declined greatly, and there were no shootdowns after 1960.