This was the case up until 1972, when the left-liberal 'progressives' took the Democratic Party away from its traditional labor-liberal populism and towards more of the single-issue, nonmaterial issues that appeal more to the upper middle class than to the party's traditional working class base.
If you want to avoid that, you probably have to avoid President Richard Nixon, or hasten his coming. If Nixon wins in 1960 or if he simply loses to Humphrey, right-populism doesn't have it's biggest showman and supporter. Without Nixon paving the way, Reagan doesn't have a chance in hell of being known for anything other than 'B-movie actor turned Goldwaterite'.