Irish Rail said "additional peak capacity" will be available on DART services this evening to alleviate overcrowding.
The company is "working to restore as much peak capacity as possible", a spokesperson said.
There was chaos this morning as passengers attempted to board crowded carriages travelling from station platforms on DART lines into the city.
But the company expects that "normal peak capacity" will be restored in full tomorrow morning. Planned talks between Iarnród Éireann and SIPTU will go ahead tomorrow, and vital "coupler checks" on trains will resume.
"While there may be some shorter trains, most will operate to planned size," the spokesperson said.
An ongoing dispute between DART drivers and Irish Rail escalated this morning, and DART drivers were no longer willing to carry out coupler checks on the trains – a process which takes about a minute.
Irish Rail admitted “severe disruption” had been caused to this morning's services.
This evening Mr Kenny said: "Additional peak capacity will be restored for this evening, and normal peak capacity will be provided tomorrow morning onwards, as the issue is addressed through agreed procedures."
He said this morning's chaos was "as a result of the refusal by drivers to undertake the checks, which verifies that an electrical connection is complete through the train when multiple DART units are connected together to form longer trains."
"This is to ensure that doors are correctly interlocking. Peak time DART services ordinarily operate as a mix of 8-carriage, 6-carriage and 4-carriage trains. However, this morning’s services were predominantly operated by 4-carriage trains, and some 2 carriage trains also.
Since last month, DART drivers have been asked to carry out the “coupling check” on the trains before they leave the depot.
Mr Kenny said: “A member of staff goes to the rear of the train, opens the door and the driver verifies that a light goes out in the cab and then checks that the train takes power.”
SIPTU were not commenting on the ongoing industrial dispute this morning.
Meanwhile, Irish Rail said: "Iarnród Éireann apologises to customers for the discomfort and inconvenience caused this morning when peak services operated with significantly reduced capacity," he added.