I don't know who started it, but I first became aware of it during the Democratic primary season in the winter of 2007-08. It was being spread by HRC partisans. Clinton now claims these were fringe people acting on their own. The alternative view is that Clinton wanted to retain plausible deniability but to launch the story in a dirty tricks, whispering campaign kind of way. Sid Blumenthal surfacing as a source lends credence to this.
The scattering of newspaper stories and headlines about a Kenyan birth, not to mention the Obama bio in the publisher's catalogue, long predate Obama getting into politics, so there is reason to believe that Obama himself was the original birther.
In any event, Trump was a latecomer to the story.
Thanks for explaining, sphinx. I thot it was HRC ‘operatives’ that she could deny, too. And I believe HRC brought it up again to try to trip Trump up. He got the last word, haha.