In the 1960s and 1970s, two brands of conservatism arose in the United States. Sometimes they were diametrically opposed. Other times, they combined and worked together in a united conservative front.
The first of these are the conservatives who embraced free-market ideology and libertarianism. In the decades after World War II, many Americans felt that the support of free enterprise was a positive force in the country, and the government should take a laissez-faire approach concerning most economic matters. To them, individual autonomy and unregulated capitalism were the bedrock of American society. As a result, they promoted limited government is most arenas. They wanted to dismantle the many government regulations and programs that resulted from the New Deal and opposed similar reforms in their own time.
The other brand of conservatives were moral conservatives. They were gravely concerned at what they saw as the erosion of the fabric of American society. Throughout the 1960s and 1970s, rapid and sweeping social changes were occurring in the country. Minorities were gaining more civil liberties, and many women were breaking away from traditional gender roles. To these conservatives, this was anathema to what they felt was the basis of American society. Shrinking birth rates, a result of increased access to birth control and the rise of the working woman, was often cited as a sign of the American decline. Unlike the more libertarian-minded conservatives, these people did not object to more government programs as long as they helped enforce traditional notions of family and society. That being said, they still often also rejected big government.
Both these styles of conservatism had some important similarities. For one thing, they were vehemently anti-socialist. As such, they saw the Communists, embodied by the Soviet Union and China, as more than just geopolitical rivals. To them, the very ideology of socialism was a threat that had to be closely watched for at home and opposed abroad. They typically advocated for taking a harder approach to waging the Cold War and rooting out socialists at home.
The mainstream Republican Party of the 1960s and 1970s would be considered moderate, even somewhat liberal by today's standards. Though one could argue that the split between the "conservative" and "liberal" wings of the Republican Party started after the Second World War, for brevity, we will stick to the era in question.
New York Governor and Vice President Nelson Rockefeller, California Governor Ronald Reagan, and former Vice President and future President Richard Nixon were alike in terms of their strong authoritarianism, which was appreciated by conservative white voters who were alarmed by uprisings in urban areas and unrest on college campuses.
On the other hand, the mainstream Republican Party was not always averse to social programs or even the protection of rights for groups toward which they might otherwise have been averse. For example, Nixon created the Environmental Protection Agency (EPA) on December 2, 1970. During his years as governor of California, Reagan helped to defeat the Briggs Initiative, a ballot proposition sponsored by conservative Orange County state legislator John Briggs that sought to ban gays and lesbians from working in public schools. Reagan formed an unlikely alliance with President Jimmy Carter and fought the initiative on the grounds that it violated free speech, given that those targeted by the ballot measure would have to inhibit themselves from saying anything about their orientations. Another rights argument was that it violated equal protection under the Fourteenth Amendment.
In New York, Rockefeller expanded the state university system. While running for the Republican nomination in 1964, he supported civil rights as well as initiatives for cleaner air and water. His opponent, Barry Goldwater, reflected the more conservative—or extreme—strain that existed within the Republican Party.
Goldwater's politics were fervently anti-Communist. He strongly opposed the social programs proposed by Democrats, whom he believed were trying to turn the United States into a socialist state. He also opposed the passage of the Civil Rights Act of 1964, believing it to be an example of the federal government's overreach into states's rights.
Goldwater's views, arguably, provided a blueprint for many members of today's loosely constructed Libertarian Party, which, like Goldwater, strongly supports gun rights and is very watchful of government interference in private and local affairs. In terms of civil rights, both Ron Paul and his son, Rand Paul, have voiced opposition to the "federal overreach" of 1960s civil rights legislation.
To summarize the two sides, one side (Rockefeller, et al.) was composed of Republicans who believed in a degree of social engineering that reflected conservative values; the other side believed that the government should not interfere with states's rights, even when it seemed to oppose practices in those states. Therefore, Goldwater probably would have opposed the interference of President Carter in the possible passage of the Briggs Amendment, thinking that such decisions should be left up to the people of the state.