In the event of an actual nuclear apocalypse, which would more than likely be the result of the cold war, a fallout shelter would be essentially useless. What made them so common?
Fallout shelters are not essentially useless. You need to keep in mind that it wasn't until the late 1960s, early 1970s that the Soviets would have been capable of devastating the US, especially if it were launched as a retaliatory strike. In the meantime, the most likely threat to any given city was a few weapons of relatively low yield and/or accuracy. It doesn't take terribly much to reasonably harden a shelter to survive even a megaton yield weapon detonating a mile or more away so the main concern is going to be outlasting the resulting radioactive fallout until it is safe to be outside (a matter of a few days most likely at this point in the Cold War). It's a fairly low cost way of majorly increasing survival rates. It's the nuclear war equivalent of an earthquake survival kit.
It should also be kept in mind that even later on, a nuclear apocalypse was by no means the only possible result of a nuclear war. A nuclear war might very well have kept itself to a few small symbolic strikes against city targets or even not a single strike against a city, but with strikes against CONUS bases which lead to fallout over civilian areas (or, as happened in one US Navy war game, a conventional strike against a nuclear power plant leading to major radioactive contamination, page 53).