Staff received 595 calls in June and July compared with 323 in the same period last year.
These are the numbers of calls that were referred to police and social services by the charity.
John Cameron, head of the helpline, said: “The number of calls we took this summer was significantly higher than last year.
“Eight months on, we can confidently say that the Savile scandal is changing the way in which people react to abuse.
“There appears to be a clear shift and the public now seem better equipped and more confident to report their concerns.”
He added: “It’s very encouraging to see that adults, including those who don’t have direct responsibility for children, take action if they think a child is at risk.
“The Savile scandal has shocked the nation but has also increased public awareness of how difficult it is for children to speak out and how crucial it is for adults to report any suspicions or concerns they have straight away.”
Savile, a former DJ, television presenter and charity fundraiser, was widely accused after his death of being a prolific sexual predator.
Police said he had abused hundreds of victims over 54 years, including at schools, hospitals and a children’s home.
The Director of Public Prosecutions, Keir Starmer, has said that Savile could have been charged over offences against at least three victims before his death in 2011.