I think if you define modern as starting in 1900 you would have to deem Franklin Roosevelt as the greatest U.S President in the modern era. He is held in high disregard by Republicans and Libertarians because he essentially instigated the era of 'big government' and programs to distribute wealth more equitably, as well as leading the war against fascism. Great leaders are seldom the most popular however, since they tend to be the most polarizing.
Kennedy still has the Camelot aura about him, but it is difficult to name any lasting legacy that was begun in his presidency.
Frankly there has been a litany of mediocrities in the job since. Nixon, Ford, Carter, Reagan, Bush 1, Clinton, Bush 2 and now Obama. I didn't include Johnson because imho he was the last U.S. President who had a firm grasp on how economies really worked, and by far was the most instrumental in implementing Civil Rights for black Americans. Vietnam destroyed his presidency, and his Great Society, which made him atleast as tragic a figure as Kennedy.
Reagan and Clinton were the most homespun and folksy, which often leads to popularity, but it would be a mistake to confuse that with competence.