The Royal Cornwall Hospital (RCH) has admitted its alarm system did not go off when thieves stole a payroll database on Monday night.
The personal details of around 10,000 hospital, primary care and partnership trusts employees were stored on it.
The health trust and the banking executives said there was little chance accounts were at risk.
An inquiry is under way to establish how the computer which contained the personal information was taken.
Angela Huxham, from the trust, said they have hired a company to ensure bank accounts were not defrauded.
Workers have been told to contact their banks, change PIN numbers and passwords and keep an eye on bank statements.
The trust said no patient information was stored on the computer which was password protected.