None that I can find regarding the actual movie*
Cape Fear is noteworthy for two reasons: Robert Mitchum’s iconic performance as terrifying psychopath Max Cady, and the shocking lack of censorship regarding the true nature of Cady’s crimes. In 1962 the walls of the Hays Office, which had presided over movie censorship since the ‘30s, were only beginning to crumble. Cape Fear marked a watershed moment for Hollywood; a few years earlier, it would have been impossible for a movie about a serial rapist and pedophile to exist without serious soft-pedaling, but this film is strikingly honest about Max Cady’s most despicable quality: he is a sexual predator of underage girls.
The only reference to actual censorship is pre-production in that
In the source novel *The Executioners**, by John D. MacDonald, Cady was a soldier court-martialed and convicted on then Lieutenant Bowden's testimony for the brutal rape of a 14-year-old girl. The censors stepped in, banned the use of the word "rape", and stated that depicting Cady as a soldier reflected adversely on U.S. military personnel.
*There were apparently no censorship issue for the US market but the UK did require some changes before providing a "rating".
Although the word "rape" was entirely removed from the script before shooting, the film still enraged the censors, who worried that "there was a continuous threat of sexual assault on a child." To accept the film, British censors required extensive editing and deleting of specific scenes. After making around 6 minutes of cuts, the film still nearly garnered a British X rating (meaning at the time, "Suitable for those aged 18 and older", not necessarily meaning there was sexually explicit or violent content).