This is one of the questions I've been studying as part of research into the Cuban Missile Crisis in Alaska. Interestingly, less than you might think.
During the 1950s and early 1960s, the perception was that a nuclear war would involve bombers and traditional air defense using fighter aircraft. This led to the construction of radar warning nets like the DEW Line and fighter airfields scattered across Alaska.
The profusion of fighter airfields was also driven by aircraft technology. The short ranges of fighter-interceptors meant you had to have large numbers in order to ensure reliable intercepts.
As technology improved both in the air (air-to-air refueling, drop tanks, better fuel efficiency) and on the ground (better command and control, better radar), the number of aircraft and bases was scaled back.
At times during the 1950s, the number of fighter aircraft in Alaska approached 200. By 1962, there were fewer than 40.
The rise of missile technology led American Air Force leaders to consider the age of the bomber dead.
With fewer aircraft around, there was less chance of an accident leading to war. Nonetheless, there were incidents. Several times, American reconnaissance aircraft flying from Alaska were shot down or intercepted by Soviet fighters.
The event closest to war came during the Cuban Missile Crisis. During the Crisis, a U-2 air sampling mission took off from Eielson Air Force base for a flight over the North Pole. These air sampling missions were common and used to detect and analyze Soviet bomb tests.
On this flight, however, aurora activity was high, and the U-2 pilot was unable to use celestial navigation to find his way home. With the stars obscured by the aurora, he took his best guess and flew south.
Unfortunately, that guess was wrong, and he ended up flying over the Soviet Union. After realizing his error, he turned east, but then ran out of gas. Fortunately, even at 60,000 feet, the U-2 is an amazing glider.
Unfortunately, the Soviet Union scrambled fighters to intercept the U-2. Alaskan Air Command scrambled fighters to escort the U-2 home. Because of the crisis, the American fighters were armed with nuclear-tipped Genie air-to-air missiles.
The Soviet fighters turned back, but had things gone differently, American and Soviet fighters could've ended up facing against each other over the Bering Strait.
How about on the ground? After all, the Strait freezes over the winter, and you can walk the 2 1/2 miles between the United States and Russia even today.
Starting in 1940, the Soviet Union banned all contact between Little Diomede and Big Diomede islands (Little Diomede is American, Big Diomede is Russian), creating what became known as the "ice curtain."
In the National Guard armory on Little Diomede, there was (maybe still is) a large sign: "If the Russians attack: Surrender!"
In 1988, the "ice curtain" melted with the start of Friendship Flights between Alaska and the Soviet Union. There is a great collection of oral histories in this document created as part of the 25th anniversary of the flights.
Was there even any strategic value to the straight, or those two islands in the middle? (Forget the names)
I know the US has some strategic bomber and fighter pilot bases up there. Good buddy worked there as a chief mechanic for a few years there.