Angry travellers have complained of long queues, with the systems problem reported in London, Amsterdam, Edinburgh, Newcastle, Denver, Atlanta, San Francisco, Toronto, Chicago, Berlin, San Diego, Las Vegas, Phoenix, Vienna, Rome and Durban.
The DAA has advised passengers to check latest flight information with their airline before travelling to Dublin Airport.
Some passengers have reported delays of up to five hours, with one in Denver complaining he had "been sat on the plane cooking" for "over two hours" and had been forced "to take my trousers off in an effort to cool down".
Responding to passengers on Twitter, the airline wrote: "We apologise to our customers for the delay and we appreciate their patience as our IT teams work to resolve this issue."
It added: "Our colleagues are doing everything possible to check in customers for their journey."
One passenger at Toronto Airport said: "I guess the check-in application is down and it's worldwide."
Passengers at San Francisco were handed a letter from British Airways explaining the delays.
It read: "At this time we are experiencing problems with the computer systems. As a result, in order to continue to check-in, in the absence of the computer system, we will be using a manual fallback process.
"Initially, this means the customer service staff need approximately 30 minutes to set the process up in order for it to work effectively and not to delay you later.
"Once we begin, check-in will be slower than normal, as information has to be recorded by hand."
Staff with clipboards were writing manual boarding passes for passengers, a delayed traveller at Seattle Airport said.
Matthew Walker had been waiting for more than two hours to board his flight back to Heathrow.
The 29-year-old financial analyst told the Press Association:
"People were lining up, some had already checked in and got through security, but others, when this thing happened, whatever it is, were stuck in the check-in queue.
"So they [the staff] have the problem that they didn't know who had already gone through the gate because all the systems literally just had a meltdown, basically."
Meanwhile, people travelling through London City Airport are facing disruptions today after a number of Black Lives Matter protesters "occupied" the runway this morning.
A number of British Airways and Cityjet flights between Dublin Airport and LCY have been cancelled or delayed over the protest.