We are often told that reason why the United States lost Vietnam was because of hippie movement (the socalled Counterculture) in the United States and W-Europa, and made the USSR believe that the West was weak, and that the USSR had agents in these anti-war hippie organisations.
We might be told that, but we might also tell those putative educators that they're wrong, or at least greatly overburdening the flexibility of the English language to express shades of meaning. Even with the protest movement and so forth in the United States, overall public opinion was much closer to strong prosecution of the war to a finish than that of withdrawal until quite late. President Nixon did not win elections in 1968 and '72 by appeasing the peace movement, let alone the counter-cultural one. I'm hard pressed to think of someone more diametrically opposed to those movements in personality, politics and bearing!
The USSR was fooling around at times with some of the organizations you describe, and certainly I doubt Moscow had anything less than schadenfreude for Western troubles. That being said, the effects of Soviet meddling, propaganda et al in the West were washed out by vastly larger internal forces of change, and it's hard to see a POD that results in less drastic cultural shifts relegating Soviet influence to much more than some footnotes.
What if for some reason that counter culture had not taken hold in the Youth of the West? How would that effected the Cold War? Vietnam? and societies in West generally?
Some kind of cultural shift was probably inevitable in both Europe and the United States thanks to the youth bulge of post-war baby booms, and the inherent differences of experience between generations.
Just because there is some cultural climate change doesn't mean it has to be quite so jarring though. In the US, if one avoids large-scale commitment to Vietnam, or the draft(or the draft does not reach the same scale), most of wind is taken out of peace-movement sails. In such a scenario, there's no or much less of a central interest for young men, especially of otherwise stable middle class origins. This does not mean that other major events and movements are gone, like the Civil Rights movement and feminism, so there will still certainly be challenges to the status quo in that respect. However, lacking Vietnam to radicalize young people and students, or to promote drug use as recreational escapism, or to give draft-age young men a lot to lose, there would not be a national impetus for quite so much disillusionment and attendant disorder. What I see as more likely is a period similar to the early Progressive era and the Roaring Twenties in equal measure. There'll be youth rebellion in the sense of flaunting visible norms, but it won't reach the point of total rejection of the past, and a more sober-minded constituency for a change in social and legal attitudes towards women, minorities, etc could promote a smoother transition without accusations (or fewer and less legitimate ones, at least) of socialism, anarchy, or rank immorality. Civil Rights and the birth control pill are still going to wrench American culture in a new direction, but a more peaceful and prosperous period ought to let it occur in less radical and more gradualist a fashion.
As for Western Europe, it seems more complicated in that the impact of various events, leaders, policy and culture differs more widely within it than within the United States. Perhaps if decolonization proceeds in a different fashion, and European institutions appear less reactionary or perhaps simply are more flexible in some areas, adjustment could be made with the times without a year like 1968. I don't know how you give European young adults and teens in those days different outlets for rebellion or perception of a stake in establishment culture, sadly. Discrediting some of the non-establishment left-wing sources of unrest might actually be simpler, if there is another major uprising in the Soviet Bloc, or better, if the Hungarian revolution is worse for the Soviets and evolves into prolonged, sporadic violence that spreads widely behind the Iron Curtain.
I think I have a few ideas that might work then, but here's a list:
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Possible Changes, not mutually exclusive:
--The French defeat or eke out something more than defeat in Indochina vis-a-vis the communists, perhaps pioneering or otherwise replicating the kind of success the British had in Malaya
--The Suez Crisis and ensuing war do not occur, most usefully because Nasser's coup falls apart, and peaceful negotiation between the UK and Egypt over Suez and other issues occurs without rancor. Avoiding the forcing of the issue caused by Egyptian rejection of the 1936 treaty might be one way.
--The Algerian War of Independence either doesn't happen, or is somehow easier on France. Avoiding near civil war in France and the alienation of Arab and Berber Algerians can't help but improve French stability as it confronts modernity. Tying this to the US and Western European culture in general would be harder though.
--De Gaulle isn't in charge in Paris for so long, or not at all. A natural extension of the previous possibility, France without DeGaulle could be a more flexible one, allowing for outlets of rebellion and gradual social changes instead of spurring radicalism as a reaction.
--The Hungarian Revolution in 1956 gets full Western attention and discredits extreme leftism because either the Suez Crisis is timed differently, or doesn't happen; or the Revolution and the Soviet response is far broader and more brutal, with the unrest in Poland at that time exploding into a similar situation. The Soviet crackdown would take much longer, and be far more visible, with more refugees escaping to the west with their accounts of communist utopia, helpfully with lots of media.
--The US phases out conscription, moving to a volunteer force with large numbers of reservists after WWII or the Korean War. This may be easier without the Korean War occurring. No draft means a weak peace movement.
--(Bear with me on this one) Nixon is elected President instead of Kennedy. This might have the effects we're looking for, not because Nixon will though the power of his irresistible charm and charisma lead the world into the
Second Victorian Era! ...but because Khrushchev may be less willing to push Nixon in some respects. Avoids the Cuban Missile Crisis most probably, and may have a few other impacts on Cuba(perhaps literally). Nixon, as President Establishment, would also be one of the few able to push civil rights in, appropriately, the sense that only Nixon can go to China. It would be in his interest, after all, to rip the carpet out from under the Southern Democrats in Congress and swipe some black votes. Given knockons and butterflies, the US could avoid Vietnam simply as a result.
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You'll note that all of these are post-war changes. A POD during WWII or especially before would wreak such massive change by 1960 that I think we'd lose the reference point to call it a counter-culture.
To clarify my points, I don't at all think that the youth bulge and attendant social changes of the era are avoidable without wash-out effects, but it just might be possible to ease things from staid 50's morality of normalcy to "Roaring Sixties"-style rebellion that matures into 80-90s yuppie ambition and intellectual tolerance. This really does require the Western World to experience sustained economic growth much as OTL to fuel and lubricate the whole process with wealth and disposable income I think, but your mileage my vary.
So have at thee! Let us go forth and crush the counterculture!