I am in my early 50's, but my memory of that period is fairly vivid. The conservative response to the counter culture of the60's was very tentative.
Barry Goldwater was the Republican nominee in 1964 against "the legend" JFK's successor Lyndon Johnson, the country perceived Kennedy and his successor Johnson as liberals, although they deepened our involvement in Viet Nam.
Johnson destroyed Goldwater in the election of 1964, the Beatles became the best rock group and first of its kind, with a counter culture theme, followed by many other musical groups. Conservatism was perceived as anti-progressive, out-of -the-mainstream, and even backwards. This pretty much remained the case till Johnson was overwhelmed by the escalation in Viet Nam, chose not to run in 1968, and the tide slowly began to turn. Nixon was elected ina close election, the counter culture still dominated, and helped drive Nixon from office. The media had it all its own way without any real challenge.
This trend carried into the 1970's, till Ronald Reagan came along, and implicitly and explicitly took the counter culture to task for being overly critical of American values and culture.
Don't forget that Nixon was re-elected in a landslide in 1972 against Mondull. Then he had his problems which put Gerald Ford in office who was not a strong conservative by any stretch. Then he pardoned Nixon which probably cost him the next election to the peanut farmer.