Pressed by presenter Matt Chorley on how the rise will still mean Wandsworth's council tax is lower than many councils across the country, Badenoch said it would still worry some families.
"I think if you suddenly have a nearly 100% increase in your council tax, you would be extremely worried. Families have budgets," she said.
"Just because on average it's lower than somewhere else doesn't mean that people are going to be able to afford it."
She refused to say what alternative rate of increase she would propose, but questioned why a 40% cut was being imposed on the council when "Andy Burnham has not found 40% of anything to cut" himself.
She also defended her record as local government minister when the Tories were in Downing Street, despite Conservative governments having slashed some council grants.
Changes to the funding formula for councils under the Fair Funding Review 2.0, will see money redistributed from some council areas to others.
After deciding to reduce their share of central government funding, the government gave five London boroughs - Kensington & Chelsea, Westminster, Wandsworth, Hammersmith & Fulham and the City of London - and Windsor and Maidenhead Council in Berkshire the power to raise council tax above the usual 5% cap for two years.
The London councils Kensington & Chelsea, Westminster and Hammersmith & Fulham are considering joining Wandsworth in using the Sustainable Communities Act to challenge the cuts.
Wandsworth Council's Tory deputy leader said on Tuesday: "Higher council tax will not be easy or welcome. We do not regard it as fair and will continue to challenge the government by every possible means."
Leader of the opposition, Labour councillor Simon Hogg, said: "It's outrageous, it is stunning. People who don't know how to run the council panicking and making you pay the price.
"This increase is unfair, it's unnecessary and we oppose it."
However, the PoliticsHome website reported, external that three Labour MPs in Wandsworth - Rosena Allin-Khan, Fleur Anderson and Marsha de Cordova - have refused to back the council's appeal to the government to stop the funding cuts.
The three Labour MPs claimed the council "was in a strong financial position" before the Tories took over at the May elections earlier this year.
In a statement when the council tax rise was announced, a spokesperson for the Ministry of Housing, Communities and Local Government said: "Local authorities decide the level of council tax they wish to set, but we are clear that in doing so they should put taxpayers first.
"We're fixing an outdated funding system so funding goes where need is greatest, while protecting councils through the transition and ensuring no area faces a cliff edge."
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