The survey indicated that less than half of British people now trust the BBC following allegations that Savile may have abused young people while working for the corporation.
Viewers and listeners were asked about the impact of the allegations against Savile, who was a major star on BBC television and radio in the 1970s and '80s.
The poll of 1,000 licence fee payers, conducted by Comres for BBC Radio Five Live, indicated that only 45% felt that the BBC was "trustworthy".
A similar survey in 2009 put the level of trust at 62%.
Almost two-thirds of British adults polled said that they believed the corporation would suffer "lasting damage to its reputation" from the scandal.
Alongside a drop in trust, respondents also indicated an erosion in the sense of pride in the BBC, dropping to 62% of adults, down from 76% in 2009.
Savile, who died last year, has been described as one of the most prolific sex offenders in recent UK history. Police are investigating more than 400 lines of inquiry and have said that there could be as many as 300 victims.
The BBC has come under fire over claims that some of the abuses may have allegedly happened on the corporation's premises.
But also, there have been questions over why a Newsnight investigation into allegations Savile was a paedophile was dropped last December, the same month in which the BBC ran tribute programmes to the late star.
BBC director general George Entwistle has ordered two independent investigations into the scandal - one is focused on the culture and practices of the BBC while Savile worked there, and the other is looking at the decision to drop the Newsnight report.
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