The grandaddy of them all, of course, was Barry Goldwater, who had a Jewish father. He was raised as a Christian by his mother, though it doesn't appear that he was especially religious. Goldwater was, by all accounts, proud of his Jewish heritage, but this aspect of his background played what seems today like a strange role in the 1964 campaign. From what I can tell, his handlers sought to play down his Jewish background, referring to his grandfather as a "Polish immigrant" (including in the speech given to nominate him.) Given that "Poland" didn't exist when the Goldwaters came to the U.S. in the mid-19th century, and that they were ethnically Jewish, not Polish, this was a bit misleading. If his family had been German-speaking residents of the Russian empire, instead of Yiddish-speaking, do you think they would have been referred to the Goldwaters as Polish immigrants?
Meanwhile, the Jewish community, near as I can tell, was equally eager to downplay Goldwater's Jewish background. Neither of my parents, for example, was even aware that Goldwater was ethnically half-Jewish, until I mentioned it to them. Rather than being proud that a grandson of Jewish immigrants was running for president, Jewish leaders were appalled at Goldwater's strong conservatism, and perhaps afraid that his background could stir anti-Semitism; I've read that Goldwater received only 10% of the Jewish vote. Of course, the fact that he was nominally Christian didn't help, but Jews had previously embraced New York Mayor LaGuardia, who had an analogous background, so I'm guessing his reception had much more to do with his political views.