http://www.cnn.com/2016/09/17/politics/hillary-clinton-birther-conspiracy/Hillary did disavow the birthers (although at that point in time, it was more about Obama being a Muslim than him being born in Kenya) in March 2008, long before the primary ended. Granted, she did throw in a weaselly disclaimer of "as far as I know," which I won't defend. So her hands aren't entirely clean - although I do find it amusingly ironic that if Trump had said something like "as far as I know" after a supposed disavowal of a bullshit attack on a rival, his fans would refuse to see anything wrong with it at all. "What are you talking about, he's encouraging the rumor by saying it's as far as he knows? That doesn't make any sense! He literally just made a factually-correct statement that it was as far as he knows! What, do you think he should have said it was as far as he
didn't know? Or that he should have pretended to be omniscient and claimed he knew beyond a shadow of a doubt? This is hilarious, you're suggesting that he meant the exact opposite of what he said! What a ridiculous liberal you are!"
Also, if we're defining "starting" the rumor as simply spreading it around, making it popular (as opposed to literally dreaming up the idea and being the first to voice it), then Trump deserves far more of the blame than Hillary and/or her supporters do. Yes, it became fairly well-known during the 2008 election, but it was still clearly fringe; something people knew about but nobody beyond a few random nuts actually believed. It wasn't until Trump ran with the idea that it rocketed into the mainstream. All of a sudden, a well-known figure had thrown his weight behind the idea. It was getting heavy media coverage. Polls were conducted asking people whether or not they thought it was true.