She outlined her concerns about the consequences of the Brexit vote in a letter to Theresa May on Wednesday.
Mrs Foster and Deputy First Minister Martin McGuinness highlighted five key priorities to the prime minister.
On Thursday, Mrs Foster said: "Brexit means Brexit, but that doesn't mean that we close our eyes to the challenges that are there."
She emphasised that Northern Ireland's border must not become a catalyst for illegal activity, nor create an incentive for those who wished to undermine the peace process.
"In the past, terrorists have used the border as a way of evading justice and that can not be the case in the future," she said.
"We need to make sure that we have strong mechanisms in place to make sure that they don't get away with that sort of activity."
In their letter, the first and deputy first ministers highlighted to Mrs May  that Northern Ireland was unique as the only part of the UK that has a border with an EU member state.
They also referred to costs, the energy market, drawdown of EU funding and treatment of the agri-food sector as other priorities.
During the referendum, Mrs Foster's Democratic Unionist Party (DUP) campaigned in favour of Brexit, while Martin McGuinness's Sinn FÃ©in wanted to stay in the EU.
When it comes to the economy, they said they wanted to "retain as far as possible the ease with which we currently trade with EU member states".
"It would be negligent of me not to point out where I believe the challenges are," Mrs Foster said.
"We're extracting ourselves from European Union and it is of course right that we identify where those challenges lie, but I fundamentally believe that there are huge opportunities."
Speaking on Thursday, Mr McGuinness said the letter was "a very important indication" that he and Mrs Foster "recognise our huge responsibility to the people we represent".
"I think we would've done a grave disservice if we hadn't articulated the differences and challenges we might face in times ahead," he said.
"This is about how we protect the interests of the people we represent and the challenges that lie ahead."
He added that it was important the he and the first minister "come together in the aftermath of the vote" in spite of having been on different sides of the referendum debate.
"My gripe isn't with Arlene Foster, my gripe is with David Cameron who called the referendum in the first place," he said.
"This is too big, too serious for us not to be joined up in relation to how we take this process forward."
Stormont's opposition parties criticised the letter, with the Ulster Unionist Party's (UUP) economy spokesperson Steve Aiken asking "whether this is a DUP u-turn on their position on the referendum".
Mrs Foster replied that the UUP's position on Brexit had been "comical".
She added: "Poor Steven Aiken has been sent out once again to be the attack dog against the executive, and frankly he comes across more as a Chihuahua."
However, the SDLP's Claire Hanna said: "The first minister has serious questions to answer over the breathtaking reverse ferret she is currently performing.
"Ms Foster has clearly realised the error of her ways in backing Brexit, and now the horse has bolted is asking the Conservative government to close the gate."
While the UK as a whole voted to leave the EU by 52% to 48%, 56% of people in Northern Ireland voted to remain.
First Minister Arlene Foster has denied that campaigning for the UK to leave the EU was a mistake.