This 1994 Washington Post interview of legendary Conservative-turned-Libertarian Barry Goldwater discusses his decades-long balking at the direction his party was heading into.

"... When you say 'radical right' today, I think of these moneymaking ventures by fellows like Pat Robertson and others who are trying to take the Republican Party away from the Republican Party, and make a religious organization out of it. If that ever happens, kiss politics goodbye."

He saw increasing donations from Evangelicals, like Pat Robertson, trying to buy greater religious influence in politics as a direct violation of what the United States stands for: individual liberty and the pursuit of happiness. The observation in the article is dated 1994, after Goldwater had refused an invitation to Nixon's funeral, whom he never forgave for lying about Watergate. However, he expressed having these views as early as the sixties.

Goldwater tried for the presidency in 1964 but was smeared as a radical and dumped in favor of Johnson. Further snubs came from his own party, trying to pin him down on his rather liberal views about individual rights,

"Having spent 37 years of my life in the military as a reservist, and never having met a gay in all of that time, and never having even talked about it in all those years, I just thought, why the hell shouldn't they serve? They're American citizens. As long as they're not doing things that are harmful to anyone else. ... So I came out for it."

Goldwater saw a religious political invasion as the death knell to the Republican party. In today's context, a lot of Republicans would agree with him, but back then it was much murkier. By the time Reagan was courting Evangelists, as /u/MaesterMagoo noted, they had been courting Republicans for at least a couple of decades.

tl;dr: Barry Goldwater noted Evangelical Christianity as toxic to Republicans as early as the sixties, by the seventies spiritually jumped ship on his party, and saw its descent into religious fervor as the death blow to the Conservatives.

EDIT: While he supported gay rights, he was also against the removal of Jim Crow laws, and wasn't above use of racial propaganda in his '64 campaign. He was an odd duck. It is important to remember that figures from the past are not worthy of unquestioning adoration; people might have touted individual rights, but seen no conflict with segregation, something unthinkable to us, today.

No until Reagan courted the Evangelical movements they tended to vote Democrat. Roe vs Wade was the first rumblings of a rightward shift as the GOP picked up the antichoice flag. The solidification f a religious right largely took place within the Reagan years (80-88) and saw major evangelical figures gaining prominence in the party notably Pat Robertson and Edwin Meese.