To add to what /u/restricteddata said, Alaska wasn't host to surface-to-surface missiles because the technology of the United States advanced so rapidly during the 1950s and there was a fear of an "atomic Pearl Harbor" that would destroy America's arsenal on the ground. At the start of the decade, regional jet bombers were cutting-edge technology. By the end, intercontinental ballistic missiles (ICBM) were being deployed in significant numbers.

From my research, the turning point in Alaska came with the successful 1957 Soviet test of an ICBM and the subsequent launch of Sputnik. At the start of that year, US Air Force documents show extensive plans to expand a military presence in Alaska. By the end of the year, those plans have been scrapped and a military drawdown begins. The withdrawal is so extensive that it triggers a mild economic recession in Alaska as construction projects end and soldiers (with their families) move out of the territory.

A great comparison can be made to the Soviet Union, which lagged behind the United States in mass-production of intercontinental missiles. While Sputnik is well known as the first satellite, it earned that honor because of political infighting within the American military and civilian rocketry programs. By the time Sputnik flew, the United States had many missiles capable of orbital (or intercontinental flight). A decision was made to mass-produce these longer-range missiles rather than expand production of intermediate-range ballistic missiles.

In the Soviet Union, this decision was not possible. Although Khrushchev boasted in the early 1960s that Soviet factories "were turning out missiles like sausages," this was an empty boast. The Soviet Union did not deploy large numbers of intercontinental missiles until the last half of the 1960s. In the meantime, it deployed significant numbers of intermediate-range missiles to fill the gap.

This included missiles deployed to Cuba, triggering the Cuban Missile Crisis. Had the Soviet Union actually been deploying large numbers of intercontinental-range missiles, there would have been no need to put missiles in Cuba.

At the same time that the Soviet Union was preparing to put missiles in Cuba, it was putting them in far eastern Chutkotka, at the easternmost tip of Asia. At a town called Gudym and a site called Anadyr-1, the Soviet Union constructed an intermediate-range ballistic missile base.

Completed and activated in 1963, it was operational for only five years. By the end of its life, the Soviet Union had enough long-range missiles that Anadyr was more trouble than it was worth.

As the Soviets learned, the maintenance and logistical problems involved with operating an IRBM base that far north were immense. There were repeated problems with permafrost heaves and transportation, and simply keeping the missiles operational was difficult.

In the United States, those problems had been foreseen and showed up in cost estimates for construction of a missile base in Alaska. In the Soviet Union, politics could not act as a check against construction — in the United States, it could and did.

In addition, construction of a missile base in Alaska was hindered by poor intelligence and overestimates of Soviet strength. Until reliable satellite imagery became available in the 1960s, the United States operated under the belief that the Soviet Union had significant numbers of missiles deployed in Siberia.

It was thought that any missiles deployed in Alaska would be vulnerable to a surprise attack and would not have time to launch before being destroyed on the ground. Missiles launched from Siberia would reach Alaska in less than 10 minutes, and in the case of a surprise attack, that was insufficient time to act.

Those factors combined to prevent the United States from deploying surface-to-surface missiles in Alaska.

Nuclear gravity bombs were introduced to Alaskan airbases starting in the mid-1950s, though removed by the mid-1970s. Several nuclear-armed defense weapons systems, like the air-to-air Genie missile, and the Nike Hercules surface-to-air missile, were deployed there. There were also tactical nuclear weapons deployed there, like the 155mm atomic artillery, though whether that was because anyone thought the Soviets might invade, or (perhaps more likely) because it would serve as a forward deployment site for these weapons should they be needed in the Far Eastern theatre, I don't know.

Separately, Alaska was where a lot of radars and other sensors were stationed, to see incoming Soviet attacks prior to their arrival at the continental US.

To my knowledge there were never any intermediate range missiles deployed there, possibly because the range would have to be pretty high to hit anything of interest (while there are several interesting targets on the far east coast of the USSR, they were already being "serviced" by sea-based forces and Allied foreign airbases in Asia). There is no advantage to putting ICBMs somewhere like that — you want ICBMs to be put in the middle of your country, far from the coasts and the enemy, so you can have time to deploy them before they are attacked.

/u/The_Alaskan could chime in with more details if he has the time, but Alaska was indeed militarized space during the Cold War, and did have nuclear deployments.