The prime minister made the remarks while distributing grants to the families of journalists who died or were injured in accidents and those affected by ill health and insolvency, at her offices on Thursday.
The authorities subsequently shut down the university on Tuesday for an indefinite period after activists of Bangladesh Chhatra League attacked protesters who demanded that VC Farzana's resignation.
But the move has done little to quell the unrest as the protesting teachers and students vowed to continue their demonstrations on the campus.
On the allegations against the VC Farzana, Hasina said, "They must prove the allegations. If they fail to do so, then everyone who has raised the allegations and those who are giving speeches will face punishment.”
"If they fail to prove the allegations then they'll receive the same punishment as the one they're accusing of committing corruption. This is what the law prescribes. I'm telling you clearly that we will take legal measures against those who make false allegations."