The two prime ministerial hopefuls in the upcoming general election were quizzed about Andrew's friendship with convicted sex offender Jeffrey Epstein, who took his own life in jail while facing sex trafficking charges.
The Queen's second son has been accused of showing a lack of empathy with Epstein's victims and a lack of remorse for his friendship with the disgraced financier.
Labour leader Mr Corbyn and his Tory counterpart Mr Johnson were initially asked if the monarchy was fit for purpose.
Mr Corbyn replied: "Needs a bit of improvement.”
Taking a more diplomatic slant, Mr Johnson said: "The institution of the monarchy is beyond reproach.”
Asked about Andrew specifically, Mr Corbyn said: "Before we discuss Prince Andrew, I think we should discuss the victims that are there because of what Epstein was doing.
"I think there are very, very serious questions that must be answered and nobody should be above the law, but the primary position ought to be the proper treatment of those people who were victims of the most appalling behaviour by apparently Epstein and many others.”
Mr Johnson said "all our sympathies" should be with the victims of Epstein, adding: "The law must certainly take its course.”
Following the Duke of York’s disastrous interview with Newsnight at the weekend, his role in public life has suffered major setbacks as multimillion-pound businesses cut ties and universities review their association with him.
Multinational bank Standard Chartered has joined KPMG in deciding not to renew its sponsorship of Andrew's Pitch@Palace scheme.
London Metropolitan University is to consider the duke's role as its patron, while a student panel at Huddersfield University has passed a motion to lobby the duke to resign as its chancellor.
What Andrew said
During the BBC interview, the duke, questioned by Newsnight presenter Emily Maitlis, twice said his relationship with Epstein had some "seriously beneficial outcomes", giving him the opportunity to meet people and prepare for a future role as a trade envoy.
He denied sleeping with Virginia Giuffre, one of Epstein's victims, on three separate occasions, twice while she was underage, and said an alleged encounter in 2001 could not have happened as he spent the day with his daughter Princess Beatrice, taking her to Pizza Express in Woking for a party.
The American accuser said the alleged encounter began with the duke sweating heavily as they danced at London nightclub Tramp.
But the duke said he had a medical condition at the time that meant he did not sweat.
He said he had no recollection of meeting Mrs Giuffre.