I don't believe anyone really ever believed it. I think it's like the Flat Earth thing. Nobody really believes it but they say they do to get attention, or to muddy the point, to cast doubt.
Trump said, back in the day, that he'd actually sent men to Hawaii to check it out, and they were finding 'amazing things! Amazing things!' We never learned what those amazing things were.
The Republicans never could get used to the Dems nominating a black man. This being the 21st century, they couldn't just say 'Hey, you don't want a colored guy for president, don't you?' So instead they concentrated on making him look 'not like us', 'not one of us', 'not our kind'. Not just a different race, they insisted he was a Muslim, he belonged to an America-hating black church. He palled around with terrorists. And he wasn't born here and not qualified to run for president.
They STILL call him a Muslim, and insist that he came to office with a plan all ready to go to destroy America.
For 2020 they'll insist all the candidates are 'socialists' and 'marxists' and 'communists'. A whole new birther movement has been started for Kamala Harris. And of course they want to send Ilhan Omar back where she came from. (Racist? That's not racist! How is that racist?)