A wide-ranging YouGov survey of more than 1,000 registered members of the party also found that two-thirds back Irish reunification.
Labour was firmly opposed to a Yes vote in the 2014 independence referendum, with former Chancellor Alastair Darling chosen to run the Better Together campaign. Party heavyweights such as Gordon Brown and Lord Robertson also made high-profile interventions supporting the union.
But the party's fortunes north of the Border have entered steep decline since the constitutional vote. At the 2016 Holyrood elections it returned just 23 MSPs, putting the party in third place behind the SNP and the Tories.
Among members living in Scotland, opposition to independence was stronger, at 60%, with 31% in favour while 9% did not know.
London and the north of England also had a majority against independence, at 41% to 36% in the capital and 44% to 40% in the north.
Elsewhere in the UK a majority of those polled narrowly backed Scottish independence, at 42% to 39% in the south outside London and 40% to 33% in the Midlands and Wales.
The poll suggests that the Labour membership has become far more radical in its views since Jeremy Corbyn became leader. Members overwhelmingly support electoral pacts with other parties at Westminster to reverse Brexit. Eight in 10 would support a coalition with the SNP, while more than half would support a deal with the Lib Dems.
Responding to the poll, SNP MSP George Adam said: “Scotland’s interests are being ignored and our voice silenced in the Brexit process.
“Little wonder then that support for independence is rising – particularly among the rank and file of the Labour party who must be despairing at the lack of leadership in standing up to the Tories right now.
“People know that independence offers the opportunity to build a fairer, more prosperous country that is no longer subject to the damaging agenda of Tory governments we didn’t vote for.
“On the eve of Boris Johnson becoming the next Prime Minister, it’s becoming ever clearer that Scotland must have the opportunity to choose a better future.”