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The statement came as the Vatican, responding to a letter from the Catholic Church in England asking whether Savile's papal knighthood could be revoked, said the honour died with the person but in hindsight "should not have been bestowed".
Police said this week that some 300 victims had come forward and that they were preparing to make arrests in a scandal that has already damaged the BBC's reputation.
Savile's nephew Roger Foster said the family had been unaware of the TV host's "darker side" and was struggling to reconcile the image of the man they loved with the allegations of abuse of young girls.
"How could the person we thought we knew and loved do such a thing?" said the statement. "We recognise that even our own despair and sadness does not compare to that felt by the victims."
Savile, a cigar-chomping former DJ who was one of the BBC's top presenters, died last year aged 84.
The scandal has destroyed the reputation of a man who had been widely admired and honoured for his charity work, and has raised troubling questions about the BBC's management and its workplace culture in the past.
Police have said Savile was "undoubtedly" one of Britain's most prolific sex offenders, while the head of the BBC's governing body has called the allegations a "tsunami of filth".
"Our thoughts and our prayers are with those who have suffered from every kind of abuse over so many years and we offer our deepest sympathy in what must have been a terrible time for all of them," the family statement to the Yorkshire Evening Post newspaper read.
Foster said he had watched "in horror" as British TV channel ITV first aired an expose "with allegations of a darker side to him that we knew nothing about".
Faced with growing public outrage at the allegations, the family decided to remove the headstone on Savile's grave and destroy it to avoid it becoming a target for vandals.
BBC Director General George Entwistle, who has been sharply criticised by politicians for his handling of the case, has admitted that the broadcaster has been damaged by the case.
The scandal has reached beyond Britain, generating attention in the United States, where Entwistle's predecessor at the BBC, Mark Thompson, is poised to take over as chief executive of the New York Times.
It has now reached the Vatican, after the head of the Roman Catholic Church in England, Archbishop of Westminster Vincent Nichols, wrote a letter.
"The archbishop has written to the competent office of the Holy See with a request to investigate if anything can be done about Savile's papal knighthood," the archbishop's spokesman said, adding that the letter was sent last week.
The papal knighthood is one of the highest honours bestowed by the pope and is reserved for lay people and the military. British media reports said Savile had been made a knight by the late Pope John Paul II in 1990 for his charity work.
Even as the investigation was still ongoing, Nichols acted in recognition of the "deep distress" suffered by any abuse victims, the spokesman said.
However, Vatican spokesman Federico Lombardi said there was no way to revoke a knighthood posthumously since there is no permanent list of people who have received it in the past.
"The honour dies with the individual," he said. "It's not possible to strike a deceased person off a list that does not exist," Lombardi said.
He added that the Vatican was "deeply saddened that a person who has been stained by such acts could in his lifetime have been proposed for an honour by the Holy See".
The Catholic Church has been hit by child abuse scandals in Europe and the United States in recent years, forcing it to pay hundreds of millions of dollars in compensation worldwide. The scandals have damaged its status as a moral arbiter.
The Savile scandal has raised questions over the celebrity culture in Britain, with critics saying that his stardom had given him a sense of impunity.
A top PR consultant in Britain, Max Clifford, said several celebrities had contacted him out of fear they might be associated with the scandal.
"In the last few days, I have had an awful lot of calls and expect to get a lot more, some from very famous people who in the 1960s and 1970s were in the middle of this music explosion in this country," he told Sky News.
"They are all saying that they were totally unaware and they themselves have never done anything remotely like Jimmy Savile. Naturally they are concerned because names are being mentioned." he added, according to Sky News.