I'm writing this from an American perspective, but I'm sure the experience of the 1960s varied wildly depending on where and how you lived.
As my father used to say, "If you remember the 60s, you weren't there." His point was that there was so much going on, so much exploration, so much tumult, that even keeping track of it all was difficult as it was happening, much less in the rear view mirror. It's important to note that he was in his early 20s as the decade hit full swing.
To him the decade was about exploration, about trying new things and pushing boundaries. To his parents (and to more conservative younger people), it was a time of unravelling, a time when the American status quo that held sway from the end of WWII was under direct assault. Those competing themes of discovery and unravelling could be tied to the mythos in a variety of ways.
Space Exploration: The Mercury program, the goal of which was to get a man into orbit, ran from '59 to '63. The Gemini program, which put men into space in '65 and '66, explored techniques that were later used in the Apollo program. Apollo manned flights were carried out from '68 through '72. Imagine what sort of weirdness American (or Soviet) astronauts could encounter in the cold dark of space, and how mysterious, horrible events broadcast live on TV would affect viewers down on earth.
Drug Use & Communal Living Exploration: The two didn't necessarily go hand in hand, but frequently you'd find the former if you found the latter. Hallucinogenic drugs as a vehicle for exposure to the mythos could be interesting, for example mythos-driven insanity manifesting as acid trip flashbacks. Communal living gave rise to all sorts of reinterpretations of social structure and family environments. The more isolated the commune, the more possibility that mythos elements could gain hold. Add a strong charismatic leader with a spiritual/religious bent and all kinds of nasty things could ensue.
The Vietnam War & the Unravelling of American Politics: I get the impression that from Gen X on, it has become increasingly difficult for Americans to understand what it was like to live through the Vietnam War era. The threat of being drafted was quite real to American men in their 20s, and it cut deeply into the social fabric. We're still saddled with the effects this war had on the Baby Boomers, as they re-fight the domestic battles of the 60s over and over again. What if the John Birch Society were really a front for cultists bent on promulgating war and carnage? What if Jane Fonda's trip to Hanoi was really the most visible effect of a cultist-driven effort to get popular entertainers to sow dissent? What if cultists in the highest levels of government, business, and cultural circles were playing both sides, with the goal of bringing the most powerful nation-state on earth to its knees?
The Unravelling of Race Barriers: To Jadasc's point, the fight in the 60s to make America truly a land for all people was seen by many who were involved as the high point of their lives. To others it was the beginning of the end, the end of an old order that kept some people in control and others in check. It doesn't take a stretch of imagination to conjure up images of the KKK being controlled by mythos forces, again with the goal of keeping humanity from moving forward, from gaining strength.
My interpretation has always been that mythos cultists feed on discord and strife, and this decade was chock full of both. I'll bet you could put together a great campaign set in the 1960s.
All I need now is an answer that allays my fears of the more advanced technology.
At the individual/small unit level, lethality didn't change much between the 1920s and 1960s. In fact, the antique .45 Thompson submachinegun was used in Vietnam, as was the M1911 pistol. Helicopters and warjets evolved considerably between the '20s and '60s, but they were still dependent on radios, radar and human eyeballs for targeting and navigation, all of which could easily be manipulated by the powers of Mythos creatures. Imagine what happens when a group of pilots starts chasing phantom bogies, for example.
When you move up the ladder to strategic assets like aircraft carriers, long-range bombers, and ICBMs, the control mechanisms are all still human. The Air Force crewman who is going insane won't turn the control key to arm the warhead. The engineer maintaining the engine of a nuclear-powered submarine might shut it down "accidentally". As @Sardathrion points out, man is still insignificant, particularly his mental powers. And in the 1960s, human minds still were required to control human-created technology.
It might only take one or two incidents of large-scale armed confrontation gone amok to convince the political leadership to take a more subtle approach. After all, we wouldn't want to frighten the populace, would we?