Lord Kinnock said Mr Corbyn made a "strategic error" in not allowing a free vote on the Article 50 bill.
In total, 52 Labour MPs - eight from Wales - rebelled against the party.
New shadow Welsh secretary Christina Rees has said the party should be united behind Mr Corbyn.
Lord Kinnock said the Labour leader should be making "strong salient arguments" about safeguarding the wellbeing of British people.
Quizzed on how he felt Mr Corbyn was dealing with Brexit, the former MP for Islwyn told BBC Wales' Sunday Politics programme: "I don't think it's being handled actually.
"What potentially could have been a serious problem for the Conservative Party because of the deep divisions in that party over Europe... has actually turned, because of the rather ineffectual handling of the issue by the leadership of the Labour Party, into a series of difficulties for the Labour Party.
"I actually think that the Labour Leader made a strategic error in not saying there should be a free vote, so that MPs could reflect the vote in the areas that they represent, and then simultaneously have focused really hard on what the conditions for our departure are to be."
Lord Kinnock, who led Labour between 1983 and 1992, praised Mr Corbyn's appointment of Sir Keir Starmer as the party's Brexit spokesman.
But he said Labour's leader should be "making strong, salient arguments himself about the need to safeguard the wellbeing of the British people, particularly those who are economically least secure".
"If you are a leader that what's you've got to do. That's your job," said Lord Kinnock, who was also a European Commissioner for nearly a decade.
Ms Rees told the same programme: "As far as I'm concerned we should be behind the leader and supporting him in anyway we can, because that is the structure of the Labour party."
Lord Kinnock also told BBC Wales that he thought a final agreement on leaving the EU could take far longer than the two years envisaged, and that "10 years was not unrealistic".
Sunday Politics Wales is on BBC One Wales at 11:00 GMT on 12 February
Jeremy Corbyn's handling of MPs' Brexit vote has turned Tory divisions over Europe into a problem for Labour, former leader Neil Kinnock has said.