RBS, NatWest, and Ulster Bank customers were affected in June 2012 after problems with a software upgrade reported BBC.
The Financial Conduct Authority fined RBS £42m, and the Prudential Regulation Authority fined the bank £14m.
RBS chairman Sir Philip Hampton said the problems “revealed unacceptable weaknesses in our systems”.
He said it “caused significant stress for many of our customers,” adding: “As I did back then, I again want to apologise to all customers in the UK and Ireland that we let down two and a half years ago.”
In 2012, RBS, 80%-owned by the government, set aside £125 million in compensation and costs because of the disruption. Sir Philip said that since then, RBS had invested hundreds of millions of pounds to improve its computer systems.
The FCA said that the IT failure had affected more than 6.5 million customers in the UK over several weeks.
“Modern banking depends on effective, reliable and resilient IT systems,” said Tracey McDermott, director of enforcement and financial crime at the FCA.
“The banks’ failures meant millions of customers were unable to carry out the banking transactions which keep businesses and people’s everyday lives moving.
“The problems arose due to failures at many levels within the RBS Group to identify and manage the risks which can flow from disruptive IT incidents and the result was that RBS customers were left exposed to these risks,” she added.
In addition, the banks applied incorrect credit and debit interest to customers’ accounts, and produced inaccurate bank statements.
Some organisations were unable to meet their payroll commitments or finalise their audited accounts, FCA said.
Meanwhile, the Bank of England regulator the Prudential Regulation Authority, which jointly investigated the computer incidents with the FCA, said that 92% of the customers affected had been retail customers.
Andrew Bailey, chief executive of the PRA, said: “The severe disruption experienced by RBS, NatWest and Ulster Bank in June and July 2012 revealed a very poor legacy of IT resilience and inadequate management of IT risks.
“It is crucial that RBS, NatWest and Ulster Bank fix the underlying problems that have been identified to avoid threatening the safety and soundness of the banks.”
The FCA regulates financial services in the UK, and the PRA regulates banks.
Richard Lloyd, executive director of consumer group Which? said that RBS should “pay the price for letting their customers down with unacceptable IT failures”.
“These multi-million pound fines rightly reflect how important it is to consumers that banks properly maintain the payments system we all rely on,” he said.
BBC Radio 4 presenter Paul Lewis tweeted: “RBS fines are just £8.60 per affected customer. Better surely to make bank give them all £100. This was a fundamental failure of banking.”