I think this significantly weakens Civil Rights. In the absence of Johnson's parliamentary ability, and the huge post-1964 Congressional majority (Kennedy would handily beat Goldwater, but not on the scale of OTL), I think you have a watered-down bill. This leads to long-run tensions between pro and anti Civil Rights Democrats, which the Republicans would exploit. On the other hand, this might well keep the South Democratic, since the Dixiecrats are still a viable political force. The Republicans don't get Southernised like OTL, so their Eastern Establishment wing remains viable too.
In the List of Presidents thread, I once came up with a hypothetical list of "Kennedy survives" Presidents:
John F. Kennedy (Democratic): 1961-1969
George Romney (Republican): 1969-1977
Howard Baker (Republican): 1977-1985
Walter Mondale (Democratic): 1985-1989
John Anderson (Republican): 1989-1993
Anne Richards (Democratic): 1993-2001
Joe Biden (Democratic): 2001-2005
Tommy Thompson (Republican): 2005-2013
Mike Beebe (Democratic): 2013-