Minister for Social Protection Joan Burton said she believes the Government should consider adopting a more permanent referendum commission.
She said the Government is awaiting the publication of the detailed judgment of the Supreme Court's ruling.
Following the passing of the referendum yesterday, Ms Burton said new legislation on adoption should be progressed as quickly as possible.
Speaking on RTÉ's Morning Ireland, Ms Burton said: "I think there's merit perhaps at looking at having a more permanent form of referendum commission if we are going to have referenda on a regular basis and that the referendum commission would be largely responsible for the information.
"I think that is something that the Government will have to sit down and consider very carefully."
Last week, the Supreme Court found that the spending of public money on the Children's Referendum campaign was not fair, impartial or equal, and that information in its leaflets and on its website did not conform to the required standard.
The draft legislation on adoption has already been published.
It will provide for the adoption of children in long-term foster care, regardless of the relationship status of their parents.
Sinn Féin Health and Children Spokesperson Caoimhghín Ó Caoláin said he does not think that the argument for a permanent referendum commission is sustainable.
Speaking on Today with Pat Kenny, Mr Ó Caoláin questioned whether such an approach was necessary or affordable in the current economic climate.
"If the commission is established early and given plenty of time to prepare to carry out its remit, I believe that it would be able to carry out the required function.
"And it would be remiss of us not to reflect on the fact that the Referendum Commission in this instance, under the chairpersonship of Justice Mary Finlay Geoghegan, carried out their responsibilities in an excellent way, played a straight game."
Minister of State Brian Hayes said he supports the idea of a permanent referendum commission or an ongoing education programme relating to constitutional issues.
Mr Hayes said that the Constitution was the people's document and there needed to be a degree of flexibility in the language used to explain constitutional changes so that people would be better able to understand what is involved.
"It seems to me that maybe some permanent referendum commission, or some ongoing education of people on these issues would be a good thing, because people stayed away because they weren't aware of the issues," Mr Hayes said.
Mr Hayes said the Government also needed to understand the No vote and there were "substantial arguments" on the No side that had to be reflected upon.
He said that in spite of the low turnout of 33.5% in Saturday's referendum, he thought that it was too early to come to a decision that Saturday voting should be abandoned.