It's something everyone knows, that the sweaty, pale, nervous Nixon lost the debate to the handsome, young, confident Kennedy. It's said that those listening on radio said Nixon won, but TV watchers, of which there were 74 million, said that Kennedy won.
But how do we know that the debates affected how people voted in November? It makes sense that it would, but I can't find polling data or even anecdotal evidence from people telling about how the debate changed their opinions of the candidates.
Political scientists are generally skeptical that gaffes, debates, conventions etc. have a lasting impact on the outcomes of presidential elections. See for instance political scientists Brendan Nyhan and John Sides. I'm not aware of any political science study that showed that a gaffe in a presidential election had a discernible impact on the eventual outcome. Presidential elections are primarily decided by the state of the economy. Additional important factors may be whether the country is at war, how long the president’s party has held the office, and which candidate is more ideologically moderate (Gelman and King, 1993, Vavreck, 2009, Hibbs, 2000, Bartels and Zaller, 2001).
However, a 2003 experimental study found that watching the Nixon-JFK debate on TV led to a more favorable evaluation of JFK but we can't tell from that whether it had a profound enough impact on voters in 1960 to actually affect the election.