Sure, there were always some Birchers helping get conservative candidates elected, but you give the organisation too much credit for doing things its rather puny membership was incapable of.
After Nixon denounced the Evil Axis of the American Nazi Party, the Ku Klux Klan and the John Birch Society (he always put emphasis on the last word of their name) during his comeback run at California's governorship in 1962, Welch's folks became persona non grata in the Republican Party. Whatever you may think of Nixon, you have to admit he was master of the smear.
Nixon and William F. Buckley probably did more to demonize those to their right than all the liberal pundits in NewSpeak and Slime combined.
There "influence" was not numbers though prior to Reagan they had a membership of approx. a million. Their influence came from their reputation. They embodied what was then considered the right wing. It wasn't until they turned on Reagan that their membership came down to about eighty thousand and they became known more for conspiracy theories than for political activities. No one remembers the YAF but most of the country has heard of the JBS.