Evergreen headline.
1964: Moderate GOP worry they're being drowned out by the John Birch Society & segregationists
1984: Moderate GOP worry they're being drowned out by right wing fundamentalists
1994 Moderate GOP worry they're being drowned out by anti-government ideologues https://twitter.com/GregJaffe/status/1260532102124232704

2002: Moderate GOP worry they're being drowned out by Islamophobic extremists
2012: Moderate GOP worry they're being drowned out by the gun rights extremists in the NRA
2016: Moderate GOP worry they're being drowned out by anti-immigrant bigots and white nationalists

Trying to think if there's a single time when the moderates in the GOP have had a significant, lasting impact on the direction the party has taken since 1964. None immediately come to mind.