What did John F. Kennedy do to help the cause of civil rights?
I would argue that John F. Kennedy did relatively little that was tangible to help the Civil Rights Movement succeed.
JFK came into office promising to eliminate racial discrimination in housing by executive order. He did not do this for two years. He was generally timid when it came to civil rights because he relied heavily on Southern Democrats for political support.
As his term went along, JFK did become more involved with civil rights. As the link below says,
Only late in his brief term did Kennedy take up the issue of civil rights...
Perhaps his most important contribution to the success of the movement was a speech he gave in June of 1963. In that speech, he said that civil rights was a moral issue and he committed himself to try to solve the problem. Of course, he had very little time left to live and never did get to follow through on his promise. When he died, no major civil rights legislation had been passed while he was president.
Overall, then, JFK did little to help the Civil Rights Movement succeed while he lived. However, his death did help because it encouraged people to support civil rights legislation on the idea that it was what JFK would have liked.
Kennedy's attempts to win the battle for civil rights were of course tempered by his desire to win reelection in 1964 and the need to keep Southerners happy until he was able to do so. During his inauguration, he had promised to try to end racial discrimination and, during his thousand days in office, appointed many African Americans to federal positions. He also laid the groundwork for Lyndon B. Johnson to get the Civil Rights Act that was dated 1963 passed in 1964, about a year after his assassination. Symbolically, Kennedy phoned Dr. Martin Luther King Jr's wife when King was jailed in Birmingham after a civil rights protest.