The UK voted in favour of a Brexit, with 52 per cent of voters backing the country to leave the European Union.
Ms Smith, who has just been given a post within Jeremy Corbyn’s shadow cabinet, said: “I think we need to accept the decision made by the British people - a referendum is a democratic way of making a decision.
“Now it’s for politicians to respect that decision, but to get the best deal for ordinary people.”
Ms Smith said she wouldn’t accept the result “being an excuse for an austerity budget” and said: “The poorest people in society shouldn’t be the ones shouldering the burden of a Brexit.
“If anyone is arguing for a re-run of the current referendum, I would ask them to consider what would happen if the side they were supporting had won by 52 per cent.
“It’s fair. I get that people are upset about it, I campaigned to remain, I understand people will be angry and upset, but the answer isn’t re-running the referendum. “Now we need to invoke Article 50, we need to see what Brexit would look like.
“The important thing is to keep the country together - we don’t want to see the young blaming the old, I don’t want Londoners and Scots blaming England and Wales.
“We are one Britain, I love Britain and I’m happy to be British, but Britain is only Britain because of the people who live here.
“My priority is the people of Britain and making sure they can pay their mortgages, get their pensions, that their jobs aren’t transferred to Frankfurt.
“And I think that’s right to be my priority.”
Ms Smith said she would not necessarily be against the idea of another, different referendum in two years’ time on the “exact deal” Britain is to get, which has been proposed.
She said: “I think that’s an idea we should explore, but I don’t want to say we will definitely support it if people felt that was undemocratic.
“When people were voting, it was very much about a vote on the principles of being in the EU and I do respect that and we will negotiate the best deal we can so people I represent don’t lose out.
“There are many opportunities, it might be we negotiate a deal that more than 52 per cent of people like.”