Both are correct and they have different meanings. "In the 1960s" is used to refer to the whole 60s decade while "in 1960" refer to the single year which in this situation is 1960. The first one "in the 1960s" can be used when you're talking about something that happened over a period of many years. read more
the women's liberation movement in the 1960s. is wrong is that in the 1960s should refer to a specific event, but the women's lib movement isn't an event. You could say: the rise of the women's liberation movement in the 1960s. or: the popularising of the women's liberation movement in the 1960s. as these would be things that happened during the 1960s. read more