What were the problems of the 1950s that the 1960s generation faced?
Ironically, the generation that came of age in the 1960s benefited from the prosperity of the 1950s. The "baby boomers" were more educated, more financially comfortable, and generally possessed of better opportunities than any previous generation. So many of the "culture wars" of the 1960s were not so much a result of the problems of the previous decade but rather a consequence of an educated generation becoming politically active and desiring to shed the social and cultural standards of the previous generation. We see this in a host of movements in the 1960s ranging from the emerging counterculture to feminism.
That said, the legacy of the 1950s was not as rosy as many believed at the time. The Civil Rights Movements that emerged in the wake of the Second World War left much work to do, and the Sixties witnessed many states, especially in the Deep South, retrenching in the face of the movement. The Sixties generation also inherited the Cold War, which not only nearly brought the nation to the brink of nuclear war (due to the Cuban Missile Crisis of 1962) but created a host of conflicts around the world (like in French Indochina/Vietnam) that American strategists found it difficult to manage.
The Vietnam War and the struggle for racial equality, as well as the cultural fault lines that characterized American society in the 1960s, were in these ways dangerous legacies of the previous decade. Indeed, given this perspective, one might question the legitimacy of defining the movements of the Sixties in opposition to the Fifties, tending instead to emphasize continuity between the two decades.