Chiefly because they saw him as soft on communism. Some thought he was a socialist, bent on taking the US down the "Fabian Freeway," but it was mostly that he was seen as weak and squishy and easy to push around in foreign affairs. Also, there was intense regional hostility. Kennedy was an Easterner with all those Harvard advisers, rather than a Western cowboy or man of the people.
It's hard to recover exactly what people did and didn't think then. The temptation is to translate it into terms that have currency now --"elitist," "wimp," etc. -- but that may not have been in the air then. But here's a sample:
In October, 1961, Ted Dealey, right wing publisher of the Dallas Morning News, angrily insulted President John F. Kennedy, whom he considered soft on Communism. "We can annihilate Russia and should make that clear to the Soviet government. The general opinion of the grass-roots thinking in this country is that you and your administration are weak sisters. We need a man on horseback to lead this nation, and many people in Texas and the Southwest think that you are riding Caroline's tricycle."
Dealey, whose father the famous plaza had been named after, actually had earned a philosophy degree from Harvard 50 years before and had been a crusader against the Klan in the 1920s.
There was definitely something regional and something age-related going on. The Catholic-Protestant thing may have been in the background. I'm not sure race or civil rights was a factor in Dallas (as it most likely was in other parts of the state and the country), except in so far as people like Dealey saw that as part of the whole Communist thing and Kennedy as weak in that area as well.
The Civil Rights issue was definitely a factor in the Deep South States. LBJ was only able to pass the Civil Rights Act because of the martyrdom of JFK.
It never would have passed had JFK remained President.
Again, IMO.