Let us remember that, into the seventies, the Republican party was considered the party of banks, business interests and industrial management; while Democrats represented the farmers and hard-hat workers. To the extent higher education contributes to intellectualism, the Republicans start with an edge.
Two things need to happen, and 1980 is a perfect starting point. Reagan needs to get out of the race; he has a heart attack, he is deemed too sick to run for president, etc. George Bush remains the leading contender and he never changes his position on the abortion issue. You have two choice-tolerant candidates (Carter and Bush) and both keep abortion out of the campaign, as the parties did in 1976. Bush will win, given Carter's weaknesses, and will win big in 1984, as inflation eases up big-time. Bush takes Nixon's advice not to get too close to the religious right; after all, he doesn't need their help. Falwell's Moral Majority goes bankrupt as in OTL, Swaggart "falls to sin" as in OTL, and Baker is busted for fraud. The religious-right movement does not penetrate politics. Rather, you have a minority of politicians in both parties who profess to be "pro-life" but the issue stays out of the mainstream.
The second thing that needs to happen is the Soviet Union needs to stay together. It may see China-like reform, but the presence of a powerful "communist" nation remains. The benefit? The Red-Scare enthusiasts have an enemy to rant against. They are the hold-outs of the McCarthy era, professing communist conspiracies. They vote for Republicans, but the GOP moguls just let them rant on the sidelines while presidents like Eisenhower, Nixon and Ford go about their jobs, concerned about the integrity of the United States.
When the USSR broke up, the communist "threat" disappeared and the Red Scare enthusiasts had their bubbles popped. The OTL Republicans had already taken in the religious right and the Red Scare zeal was turned away from a nonexistent communist conspiracy to a very real candidate and President Clinton. Isn't this about when the right-wing radio commentators started their followings?
The Tea Party conservatives treat their doctrine as inerrant as Holy Scripture, not subject to negotiation or compromise. Gingrich (1995) and Boehner (today) take their obstructionist stands as righteous Americans against errant presidents. To prevent this from happening, the Republicans needed to keep the social extremists from writing their platform decades ago.