The Republicans would become the party of the urban middle and upper classes as well as the intelligentsia (the former hippie radicals of the 60's, now all grown up, realising that it's the Republicans who are pushing progressive reform on healthcare, the environment and so on) while the Democrats would be the blue collar Party of the People.
One should mention that Nixon played perhaps the largest role in turning the Democrats into "elitist socialist out-of-touch pro-commie eggheads". Amusingly, Republicans won college educated voters until the last twenty odd years.
Nixon was the guy, more than anyone, that started the grand shift of blue collar voters into Republican ranks. They had been uneasy with Stevenson, and Catholic Kennedy, but Johnson had kept them on board.
He helped cement the South, but that had been trending Republican, and Goldwater kept the Rocky Mountain libertarians loyal for the next forty years but Nixon (coupled with the Democrats melt-down) brought in the blue collar voters that gave Reagan his majorities (despite them not agreeing with most of his policies).
I outline below ways both the Republican and Democratic Parties might end up, but the Democrats aren't winning back blue collar voters without shifting sharply right on economics (too much money to the poor/minorities) and social issues (too much time spent on rights for the poor/minorities).
One important thing to remember is that Watergate trashed the Republican Party.
Yes, it was growing more conservative; yes, it was winning the South and inheriting those problems. However Watergate killed old-line Republicanism. Not the Roosevelt/Progressive/Eastern/Establishment wing of the Republican Party—that finally lost in '64[1]—but the Dole/Ford Midwest "balance the budget" conservatives.
After that they were sidelined so much so that Ford, the most conservative Republican President since Calvin Coolidge, was disliked because he wasn't "conservative enough".
With a non-scandal Nixon the conservative wing of the party retains an equal footing with both the new wings: the neoconservatives and the religious conservatives of the South. Combined with the Goldwater-won libertarians of the Rocky Mountain states whose sympathies are with the Midwest conservatives that leaves the Republican Party with the same tent—but a more balanced one.
Notably supply-side economics conservatives and anti-government conservatives will be weaker, while neoconservatives will still have to fight both the Taft isolationists and the Kissinger realists instead of basically beating them both into the ground.
This leaves the Republican Party with a very different flavour. President Reagan or Ford (or at least their nominations) but Ford would be more acceptable and Reagan would probably lean more to balancing the budget over supply-side tax cuts.
At the same time the Democrats may go off in any of several directions. A neoliberal Democratic Party delivering social welfare though non-government programs (negative income tax, say) would attract those Republican libertarians. Perhaps if Daniel Patrick Moynihan gets a Presidential nomination or Presidency. Despite being socially liberal, he'd be attractive to "values voters" because of the way he framed issues that would be important—black crime because of family breakdown, in a widely attacked but agreed later on to be correct paper he did in the '60s.
A strongly anti-communist Democratic Party (if, say, Scoop Jackson got a nomination at some point) would retain a bunch of neoconservatives that left IOTL, and probably hang on better in the South (Carter/Clinton performance, instead of Dukakis performance).
(Remember that post-'68 the Democratic Party was broken just as the Republicans were broken after Watergate. IOTL McGovern/New Leftists seized temporary control until the old-line "New Dealers" reasserted some sort of control. In the ATL it is entirely possible for someone else to lead the Democrats in a new direction…*or at least an altered one.)
A key point on no Watergate is that faith in government officials, and more importantly in Washington, won't collapse. Yes Viet Nam + Great Society + Civil Rights/black riots hammered it, but Watergate finished it off. No Watergate means the anti-Washington/anti-government people don't have quite the cachet, and a broader pro-reform (in both parties) movement probably does.
So one can run against Washington in the sense of reforming it, but running against it as something to be demolished is harder.
One more point in this disjointed post: the press. IOTL Watergate made the press think they were special in a different way, just as Nixon demolished them in the eyes of a great deal of the public. Watergate confirmed to liberals that the press was right, just as it confirmed to conservatives that the press was wrong.
A press not forced through Watergate will still be hobbled by false objectivity (as that was, and is, endemic), but at the same time the press will retain (I believe) a greater sense of legitimacy.
[1] For fifty years after the 1912 Republican/Progressive split the Progressives showed up for Presidential elections, demanded their moderate-to-liberal candidate, lost (and when a conservative like Hoover won, the Republican Party was dead for a couple decades proving the Progressives point), and then went away for four years.
(Technically it was Calvin Coolidge that screwed up, but conservative Herbert Hoover did try to do something… he was just so bad at press relations it never appeared that he was doing something to help.)
This led to a mostly conservative Republican Party, especially in the House, with a big influx of moderates and liberals in the Senate and every four years for the Presidential race.
Couple that with a locked out South they had to win 2/3 of all other electoral votes to take the Presidency—which they did consistently, pre-FDR, losing only to Grover and to Wilson because Taft/Roosevelt split the party—the Republican Party was heavily conservative most of the time but starting with the election of Teddy Roosevelt they stopped getting what they wanted because of those darn Progressives coming back every four years.
Conservative activists woke up in the late '50s/early '60s, conducted a hostile takeover of the party, ran Goldwater and broke the liberal Republicans back in 1964 with the defeat of Rockefeller.
Every liberal Republican after that was just lingering on, though moderates would survive in the Northern states until now, more or less.