Losses from online banking fraud rose by 48% in 2014 compared with 2013 as consumers increasingly conducted their financial affairs on the internet.
The rise is due to increased use of computer malware and con-artists tricking consumers out of personal details, Financial Fraud Action said.
In addition, fraudsters are targeting firms in order to steal bigger amounts.
However, the £60m loss is considered to be "relatively modest" with more than half of UK adults using online banking.
Overall losses on UK cards from fraud totalled £479m in 2014, up 6% on 2013, according to Financial Fraud Action.
Anyone who is the victim of fraud on their cards is refunded unless it is proved they have been negligent.
Campaigners say the figures prove banks should continue offering choice to customers who might want to go to a branch.
"Many banks and service providers want to encourage people to manage their accounts online and will stress convenience and speed as selling points," said Judith Donovan, who chairs the Keep Me Posted campaign.
"However, the fact remains that online fraud is increasing year-on-year with many criminals having a demonstrably greater grasp on technology than many of the institutions they are targeting.
"This is particularly concerning for older or vulnerable people who might not be as capable when using technology - how can these people be sure that they are not being targeted by criminals?"
The total amount of fraud is down 21% from the peak of £609.9m in 2008.
The action group said that banks and card providers had tightened up their security features. Fraudsters have now shifted their attention to tricking people out of their personal details with scams and tall stories on the phone.
It is calling for a nationwide campaign to raise awareness of deception crimes. It has also encouraged people banking online to ensure they have the latest anti-virus software installed - which may be available free of charge from their bank.
The figures also showed that losses caused by criminals using UK cards fraudulently abroad, where they can circumvent some security features, were up sharply. Losses increased to £150.3m in 2014, up 23% from the previous year.
The figures come in the same week as fraud prevention service Cifas said that 46-year-old men were the most likely victims of identity theft.
They also come on the day that NatWest admitted some of its customers were not seeing money transferred, owing to "system issues".