As Reagan put it, "I didn't leave the Democratic Party, the party left me," so for him to stay, the Democrats would presumably not be as liberally-dominated as they were in real life, which would have a big effect on partisan political alignments. The approach which Scoop Jackson was in favor of - domestic moderate-to-liberal politics, hawkishness abroad - would likely be much more mainstream (if not even dominant) within the Democratic Party.
While the Democratic Party had undergone a LOT of changes between FDR and Reagan's rise to prominence, I do think that some of that quote was a bit of political acting on Reagan's part - he was a lot more liberal in rhetoric during his younger days, trashing basic tenants of conservative ideology that he would later champion.
