Probably the people who suffered from the Recession of 1975 would describe it as the most important problem. Coming after the oil embargo by OPEC after the American government decided to resupply the Israel military during the Yom Kippur wars, an embargo which created an oil shortage in the country, as well as driving gasoline prices up, the economic recession put many people out of work after numerous companies and huge apartment complexes and developments went bankrupt. And, with double digit-inflation during the Carter presidency, the economy was in a shambles.
From a cultural historical perspective, it may be argued that the most important ccultural problem was the counterculture movements epitomized by the Hippies. While the Hippies had various objectives, including the freedom of unrestrained expression and inner self-fulfillment ("Tell it like is" and "Be sure to wear some flowers in your hair."), an unintended spin-off counterculture movement was the drug scene and the "happenings" that drug experimentation (then, it was experimentation) engendered. The consequences of these cultural problems have not left us yet, and one might argue they have only escalated over time.
In the early 1970's I would have to say that Vietnam, namely getting out of Vietnam and ending this unpopular military engagement, was a significant crisis for Nixon and Ford. The bad feelings about this war in regards to the veterans and the citizen's attitudes toward the government in general had a negative aftermath that was felt for many years.
I would agree with Pohnpei. The staggering rate of unemployment along with the economic conditions--inflation, energy crisis, and etc.--was the biggest crisis of the 1970s. Yes people had less confidence in their government, but the economic issue is what hit hardest in most households.
President Carter called it a "Crisis of Confidence". Americans lost faith in government through Vietnam and Watergate. They lost faith in the economy through layoffs, stagflation and an obvious oil dependency, and our inability to obtain the freedom of our hostages in Iran, even after a year of negotiations. We felt weak, vulnerable, and impotent. This affected everything else in the country from investments in the stock market to consumer spending, and contributed to the larger problems we had in the 1970s.
The major problem that the US faced in the 1970s was economic. This was the issue of "stagflation." Stagflation is an economic problem in which there is both high inflation and high unemployment.
Stagflation came about because of a variety of factors. One of the most important of these factors was the huge government spending that was left over from the Johnson years. The spending on "Great Society" programs and the Vietnam War was a major cause of the stagflation. The other major cause of the stagflation was oil. Oil prices jumped and supplies fell during two crises in the '70s. Between them, oil prices and the out of control government spending caused stagflation.