And she said that despite rumours about the 'Top Of The Pops' and 'Jim'll Fix It' presenter, colleagues at the BBC were too frightened to speak out.
A documentary due to air tomorrow presents testimony from a number of women who allege Mr Savile sexually abused them in his Rolls-Royce and in the BBC Television Centre in London when they were schoolgirls during the 1970s.
Ms Rantzen believes it has taken until now for the allegations to surface because of the climate at the time.
"What child had the right to make an allegation against someone who had that sort of position in this community?" she said. "Just as with priests in their own community, they were above suspicion.
"So it was with Jimmy. He did tireless charity work, raising millions of pounds, he knew Margaret Thatcher and he stayed at Chequers. He knew the Princess of Wales -- he had long conversations with her."
Ms Rantzen, the founder of British child protection charity Childline, is predicting more victims will come forward with the emergence of allegations against Mr Savile, who died last year at the age of 84.
She said that for years there had been persistent rumours across the British media about Mr Savile's inappropriate behaviour towards minors.
"I met him half-a-dozen times. I worked on 'Jim'll Fix It' maybe once or twice. I can't say I warmed to him," she said.
"Of course I heard the rumours, but rumours without evidence are just gossip."
Ms Rantzen, who took part in the documentary on Mr Savile, 'Exposure: The Other Side of Jimmy Savile', due to air on ITV1 tomorrow evening, has no doubt about the allegations.
"Having seen the sensible, factual statements by these women, I'm convinced they're telling the truth," she said.
Paul Gambaccini said his former Radio 1 colleague was "about to be exposed" by one paper, but to prevent its publication he gave an interview to a rival which had the effect of stopping the negative piece.
ITV last night defended the documentary after a relative of Mr Savile said his family were "disgusted and disappointed" that the allegations were being made when he was no longer around to defend himself.
Ms Rantzen agreed it would have been better to air the allegations during Mr Savile's lifetime, but added: "We cannot continue the conspiracy of silence that existed during his lifetime. As long as it remains an agonising secret, you can't find healing."