The major "evidence" that I have seen for this claim has to do with the idea that the Soviet Union was not a serious threat to the United States during this time. This argument holds that the US had something in the area of 5,000 nuclear warheads that it could deliver. In contrast, the Soviets had no more than 300 and probably much fewer than that.
That is the evidence. The argument that derives from that evidence is that the Soviets would never have attacked the US given this discrepancy. The Soviets would have known that any nuclear attack by them would have resulted in a tremendous response by the US that would have destroyed their entire country. Therefore, the argument goes, there was never any danger that the Soviets would really have attacked. This means that there was no real danger from the Soviet missiles.