There's a huge amount of potential butterflies here, but I'll try to keep it concise. Assuming that the Kennedy political dynasty still comes into existence, it probably doesn't look the same as it does today in many respects. As a matter of fact, John F. Kennedy as we know him probably isn't the same person, quite literally. His grandfather, P.J. Kennedy met his grandmother in Boston. So if we assume that the question here is "What happens to the US without the Kennedy family?", that makes it a bit easier to answer.
Joe Kennedy, JFK's father, probably never exists either (or, again, is a totally different person, same in name only). Which means he's never SEC Chairman under President Franklin Roosevelt, and never the Ambassador to the United Kingdom in 1938. Joe Kennedy believed that we could compromise with Adolf Hitler, and held out on that belief for a while. That belief, combined with his eventual comment that Democracy was "finished" in England, and potentially in the US, diminished what previous political influence he had. Without him, maybe we enter World War 2 earlier, but I'm doubtful on that point. On the whole, no Joe Kennedy mostly has impact on the financial world, but I have to imagine another ruthless businessman would fill his shoes.
No JFK is a bit more significant. The 1960 Democratic Nominee might be Hubert Humphrey, or Lyndon B. Johnson instead (or perhaps some combination thereof). Could one of these two beat Nixon? Possibly, but given how close the election was IOTL, and how uninspiring these two are (at least compared to JFK), I might give it to Nixon. I don't think either of these men would do as well in the televised debates as JFK did, nor do I think either (especially Johnson) would have the same huge support of African-Americans. The Democratic Party looks different as a result, though what direction it goes in is hard to say. But for sure, they lose out on a number of liberal icons. Without the Southern Strategy on part of Nixon, the traditional party lines remain intact for at least a bit longer. How much longer though, i'm not sure.