JFK is a big reason our immigration system is as f'd up as it is today.
I would assign considerably more blame to JFK's little brother Ted, who heavily promoted the
Immigration and Nationality Act of 1965 (the Hart–Celler Act), which was signed into law by Lyndon Johnson. During debate on the Senate floor, Senator Kennedy, speaking of the effects of the act, said, "our cities will not be flooded with a million immigrants annually. ... Secondly, the ethnic mix of this country will not be upset".
I guess he figured that we'll drive off that bridge when we come to it.