The New Deal Coalition died in 1968. By 1972, the fringe was in enough control to cause trouble, and the Democrats had stopped being the majority party of the US.
1) The radical leftists which earlier spurned the Democratic Party decided to try to take it over instead. Despite FDR's reputation on the left, the New Deal Coalition was very different than what the SDS/red-diaper babies wanted it to be.
2) They rigged the rules so that Democratic Convention delegates were no longer dependent on actually being elected, but instead had to reflect a certain pre-determined diveristy. Thus in 1972 the Chicago delegation led by Mayor Daley was refused, and an entire new one was appointed in its place. While this lead to a undoubtedly more "diverse" delegation, it cannot be said that it actually represented the people in Chicago better in any way.
3) The failures of the LBJ Presidency discredited the New Deal Coalition. Not only was the war in Vietnam wearing everyone down, but the Great Society programs clearly failed to end poverty and only seemed to worsen them. Advances in civil rights seemed to only encourage race riots. Crime was escalating. The policies of the New Deal Coalition no longer seemed to work. This alienated voters, and demoralized much of the old guard that they agreed to let the left wing change the Democratic Party, which only alienated voters even more.
It was very common to hear people in the 1970s say "I didn't leave the Democratic Party, the Democratic Party left me."
The only reason the Democrats kept control of Congress for years afterwards was 1) the Watergate scandal, 2) many old time New Deal Democrats kept getting elected in their old burroughs, 3) the South was still Democratic in name and kept electing the same people they always did to Congress even though they voted Republican for President (much of the Southern Democrats often voted with the Republicans anyway).
A lot of bottled up change was happening in the 1960s. You had racial issues, you had feminism, the sexual revolution, and others. The Vietnam War was only going to make it worse. The system just couldn't handle it anymore, and once it seemed like the government couldn't hold back anarchy, a political realignment was inevitable.
You have several different options to mix and play with.
1) Somehow keep the New Left out of the Democratic Party.
2) Prevent the Vietnam War or keep the US involvement to a minimum. Maybe instead of Westmoreland being put in charge in the beginning, Creighton Abrams was. The Vietnam protests pushed college enrollment, and lots of unemployed young men always leads to trouble if they are dissatisfied about their prospects.
3) Don't have the birth control pill be invented until later. Political feminism not associated with the sexual revolution would have a very different impact. Not sure if this is really feasible. A sexual revolution that came after other issues had been dealt with might prove less traumatic.
4) A completely changed Great Society. There were a lot of flaws in the War on Poverty. A lot of this was due that LBJ was skilled enough and had a good congressional majority that he could shove through programs without too much debate and changes.
There are probably more that do not come to mind. You can't do anything about the racial issues in the 1960s. The civil rights era had arrived, and people were going to have to deal with all the changes it created, even the unintended ones.
The two critical areas are probably avoiding the Vietnam War and preventing LBJ from being so dominant. It probably means that JFK has to live which is ironic since the Kennedy''s represented the conservative wing of the Democratic Party (despite the mythology that now surrounds JFK and RFK).