Sir Keir was speaking on a visit to Batley to meet Labour’s Kim Leadbeater, the party’s candidate in the Batley and Spen Parliamentary by-election on July 1.
The Labour leader said the Government should admit its mistakes if Covid-19 restrictions were not lifted as planned on June 21.
Speaking at Batley Bulldogs’ Fox’s Biscuits Stadium, Sir Keir said: “I don’t agree with local lockdowns and where there are high levels of infection - and obviously Kirklees is one of them - people have been let down by the Government.
“The Delta variant got in because we didn’t have a strong enough border policy. There were delays in shutting down the border, particularly in relation to India, for reasons that everybody understands in relation to the Prime Minister’s proposed trip there.
“More generally we haven’t had a robust borders policy pretty well throughout the pandemic. It’s people in Kirklees who are paying the price for that.
“I don’t want people in Kirklees to go through more agony of local lockdowns because the Government has got this wrong.
“I call on the Government to take more robust measures, even at this late stage. They should level with the people of Kirklees and elsewhere that they bear responsibility.
"They bear responsibility if we don’t unlock on June 21 and there’s one reason for that - the Government left the back door open.”