The class contrast over social issues has always been there but was deepened by the social developments during the 60s. To prevent the rise of right-wing populists playing on blue-collar insecurities about inner-city riots and hippies, you'd have to either:
a) Do away with most of the divisive cultural issues of the 60s, by preventing involvement in Vietnam and directing all that money towards fighting poverty in the ghettos (hopefully thereby butterflying the rioting). If these cultural issues are introduced into the American mainstream more gradually, without the apocalyptic and violent divides of that decade, working-class Americans may be more willing to accept them.
b) Have the left keep a eye on economic issues during this period. At the time, right-wingers such as Nixon were willing to be economically centrist or liberal while saving the conservative rhetoric for thrashing the hippies - thereby winning over the blue-collar vote. Liberals, by contrast, were seen as spending most of their energy on Vietnam and civil rights. (The term "limousine liberal" was coined to describe the 60s-era mayor of NYC, John Lindsay, who was seen as very soft on crime and unconcerned about the working white people whose property was destroyed by rioters). If the "movement" of the 60s is a movement for economic justice as well as social change, the two may become associated in the mind of the public.
In the TL I've been working on for a while, a serious economic downturn begins in 1968, and the New Left and New Politics begin to focus a lot of rhetoric on the massive military spending going on and pointing out that the Nixon administration is "robbing hardworking Americans" during a recession. This is coupled with an anti-veteran backlash by conservative thinkers (long story, but it makes sense in context), which alienates patriotic working people, and a strategy by the Socialist Workers Party (who have a lot more resources for various in-TL reasons) to infiltrate the UAW, which already had a lot of young radical members in OTL.
All of that combines to cause lower-class Americans to take the movement of the 60s a lot more seriously, and stunts the efficacy of right-wing fearmongering on social issues.