The survey revealed more than 60 percent of people living in the north, and more than 80 percent of those living south of the border, would want the process to be scrapped said if a deal was reached between the UK and the EU which included a hard border.
The survey, jointly commissioned by RTE and BBC Northern Ireland, also showed that 62 percent of those living in Northern Ireland believed a united Ireland was more likely as a result of Brexit.
The figure was significantly higher than in the Republic, where the figure was 35 percent.
As Brexit talks reach the end-game, the two television networks teamed up for a special live Brexit programme on Monday night.
Attitudes about Brexit and its effect on people on both sides of the border were explored.
RTE’s Claire Byrne and the BBC’s Stephen Nolan swapped studios, with Ms Byrne broadcasting from BBC Northern Ireland in Belfast and Mr Nolan going live from RTE studios in Dublin, both in front of live audiences.
In Belfast, Fine Gael Senator Neale Richmond, journalist Susan McKay and Ben Lowry of the Belfast News Letter joined Ms Byrne.
In Dublin, Mr Nolan was joined by the leader of Traditional Unionist Voice Jim Allister, Sinn Fein’s David Cullinane, Alliance Party leader Naomi Long and Conservative MP Chris Chope.
Asked how they believed Brexit would affect their own financial situation, in Northern Ireland 55 percent said they thought they would end up worse off, while just nine percent said they thought they would be better off, and 30 percent said it would make no difference.
In the Republic, 34 percent of respondents felt they would be worse off, with two percent believing they would be better off and almost half said it would make no difference.
The poll was carried out by Amarach Research in the Republic and LucidTalk in Northern Ireland.
More than 2,000 people were interviewed across the island last week.
The question of arrangements for the Irish border is central to the Brexit debate.
Brexiteers have accused the EU and the Irish Government of focusing on the issue as a tactic aimed at gaining leverage of Prime Minister Theresa May.
However, others have suggested Britain glosses over the problem at its peril.
Speaking in October, Ireland’s Taoiseach or Prime Minister Leo Varadkar warned of a “very real” risk of a return to the dark days of the Troubles in Northern Ireland if a hard border returns.
He added: “I just wanted to make sure that there was no sense in the room that in any way anybody in Ireland or in the Irish government was exaggerating the real risk of a return to violence in Ireland.
“The two parties representing the two communities in Northern Ireland have been unable to come together to form a coalition government, which they have been able to do for most of the last 20 years.
“I met both leaders of the two main parties and they were both in agreement that the uncertainty around Brexit was one of the major reasons why they haven’t been able to form an Executive, so we can see the uncertainty of Brexit is already having an impact, if we were to have a hard Brexit you can imagine the effects that could potentially follow.”
The previous month, Dr Kate Hayward, a reader in Sociology at Queens University Belfast, told Express.co.uk: “People have forgotten what it used to be like.