There were too many 'crisis' events to list during the 1960's that were directly caused and/or impacted by the Cold War.
The Bay of Pigs embarrassed our government and emboldened Cuba/USSR. In 1962, the Cuban Missile Crisis saw the first drills for children during a nuclear strike scenario. It scared and perhaps traumatized millions. The threat of Nuclear annihilation was never as high. It also tilted the scales somewhat when the USSR eventually balked and the crisis was diffused.
Vietnam started ramping up after the JFK assassination. This was a Cold War 'Hot War'. Many other wars, most notably the 1967 War between Israel and 3 of its neighboring enemies, was a purely proxy war between the US and the USSR.
The Landing on the Moon was a blow to the USSR and started a space race.
All is all, the 1960's saw escalations of the Cold War and many dramatic events that illustrated it's terrible possible outcomes.