After the first week of the campaign, a prominent protestant leader, Norman Vincent Peale, claimed that a Catholic president would be obliged to obey the Pope or else be a hypocrit. Following this, Kennedy felt that the upcoming campaign would not be between a Republican and a Democrat, but rather between a Protestant and a Catholic, and he thought it was going to be a very hard battle to win.

Therefore, he addressed the Catholic issue shortly afterwards at a meeting of several hundred Protestant priests in Houston (speech can be found here: http://www.jfklibrary.org/Asset-Viewer/ALL6YEBJMEKYGMCntnSCvg.aspx). The general gist of his speech is that he would not take kindly to any influence the Catholic church might try to exert on his decision-making and would not answer to any demands. After this, the Catholicism issue became less pertinent in political debate and the focus shifted to other issues.

As an interesting sidenote, Catholic leaders had mixed feelings about his candidature and many voted for Nixon, as they feared a Catholic president would lead to increased religious tensions and sectarian violence.

Source: Geoffrey Perret, Jack: A life like no other

I think that once he won a heavily Protestant state like West Virginia, the American media realized that his religion probably wasn't going to be a big deal. I do know that many fringe Protestant radio hosts tried to dissuade Americans from voting for a Catholic, but it kind of backfired. American Catholics, especially on the East Coast, who were originally going to vote Republican began to suddenly rally behind Kennedy as these anti-Catholic insults began to pick up. Also some American Protestants and Republicans started getting embarrassed by their claims that he was going to be controlled by the Pope, and began to distance themselves from their cringe-worthiness. So I think Catholics played a big part in him picking up votes, but he couldn't win the election through a minority vote alone. He really needed LBJ on his team as well to pick up Southern States that the Democrats desperately needed. And let's be honest, it'd be hard to argue that JFK was truly the underdog. Sure Nixon was recently the Vice President, but Eisenhower really didn't leave the Republicans with a guaranteed landslide win, and at some points Nixon had to distance himself from Ike to prevent more campaign damage. JFK may have been Irish-Catholic; but he came from a huge family of extremely wealthy Irish-Catholics with extensive contacts with the media, the East Coast elites, many American businesses, and even some mobsters. Compared to Nixon's impoverished upbringing, JFK might as well have been aristocracy.