Police said the man was perceived to be Caucasian in appearance, in his mid-to-late 20s, about 175cm tall and has short black hair.
He was wearing a black jacket, black pants and blue face mask. He was also in possession of what appears to be a hacksaw.
Anyone with information is urged to contact Australia's Crime Stoppers.
The City of Yarra spent a fifth of a NZ$146,000 taxpayer-funded grant from the TAC on the artwork – dubbed "Fallen Fruit" – in Rose St, Fitzroy.
A council spokeswoman said the sculpture, installed in early November, "activates" the area, encourages increased pedestrian activity and had been "well received by residents and visitors".
A 300-word description of the artwork published on the council's website says it attempts to "subvert" the 1970s phenomenon of oversized, kitsch roadside objects.
"The work does this by employing the symbol of the banana, anthropomorphised through the inclusion of a human skull, a memento mori to meditate on our Western tendencies towards unsustainable desires and excess," the description reads.
"Using absurdity and humour as an entry point, this oversized pop object reveals the 'infallibility' of the superhuman figure as social myth."
Artist Adam Stone said the purpose of his banana sculpture was to represent the overconfidence in western society's approach to the climate and finite natural resources.
"This is a very small budget compared to what councils spend on other projects and consulting for those projects," he told the Herald Sun.
"The production budget provided a public artwork as well as employment for a professional painter, artist's assistant, engineer and local concrete contractor, all of whom were sole traders.
"My ambition was to create a sculpture people could enjoy of all different backgrounds."
