In what ways did the African American civil rights movement influence the protest methods of other groups in American society?
The use of non-violent public protests and the advocacy of legal changes by the Civil Rights movement of the 1950s and 60s influenced the protest methods of other groups in American society. The Civil Rights movement pioneered the use of non-violent techniques, inspired by Gandhi and used by Martin Luther King's Southern Christian Leadership Conference (SCLC) and other groups. These techniques included public protests, sit-ins, boycotts, and other forms of non-violence. In addition, the Civil Rights movement pushed for new federal laws to guarantee civil rights and won the passage of the Civil Rights Act of 1964 and the Voting Rights Act of 1965.
The women's rights movement, the lesbian and gay movement (now the LGBTQ movement), and the Native American rights movement, among others, employed these techniques. For example, the women's movement held many peaceful protests and supported the Equal Rights Amendment to the Constitution (which was never passed) to create legal and economic equity for women, just as the Civil Rights movement had pushed for the Civil Rights and Voting Rights acts.
In 1965, two years after the 1963 March on Washington by the Civil Rights movement, the first demonstrations for gay and lesbian rights were held outside the White House. Gay rights activists later campaigned for equality under the law, including the right to join the military and later to marry. In 1968, AIM (the American Indian Movement) was formed and held demonstrations, such as the 1971 takeover of Alcatraz, to protest the abrogation of their treaties and other human rights and economic issues that affected them. All of these movements build on the non-violent protests and agitation for legal protection that the Civil Rights movement developed.
I think this is a great question. I have been fascinated with this topic for so many years. Indeed, the Civil Rights Movement laid the fundamental groundwork for other movements that followed. Women's movements in the late 1960s and afterwards appealed to the social dynamic of being heard, borrowing language and syntax from the Civil Rights Movement. The placard which read, "I am a Man" from the 1950s marches throughout the South was appropriated by the Feminist movement and even through song with Helen Reddy's "I am Woman." The workers' plights were articulated by leaders like Cesar Chavez, whose hunger strikes and methods of protest were reminiscent of Dr. King. Finally, the notion of acceptance and tolerance, the idea of waiting too long and demanding entrance was a recurrent theme in Civil Rights action seen in the thoughts of Malcolm X and Kwame Ture. This very idea "came out" with the Gay Rights Activism of the late 1970s. The very idea of "out of the closet, into the street" is very reminiscent of the more active Civil Rights Activist. The issue of representation and the crisis in the lack of it, first discussed by African American thinkers in the Civil Rights Movements, became something that was brought out by other movements that followed.
Since the Civil Rights Movement for black rights was far and away the most successful movement in the last century (the movement for women's votes took a lot longer to win its goals), it is not surprising that its methods influenced everything that has come after it.
Perhaps the most obvious influence is the use of sit-ins where the protestors attempt to force the police to arrest them. This has been used by movements that range from anti-abortion protestors to anti-nuclear protestors. The use of nonviolent civil disobedience has also had an effect on the ways in which most other protest groups have tried to get their messages across.
Another thing that the Civil Rights Movement did was to craft protests that would get media attention and thereby publicize the cause. This impacted other movements as well. You have/had things like women burning their bras, gay pride parades with people dressed (or not dressed) in very "out there" ways, and abortion protestors displaying pictures of aborted fetuses. All of these are meant to attract attention just as the marches of the Civil Rights Movement did.