Labour are demanding to know whether anyone posing a risk to national security was allowed to enter the UK during a period in which border controls were allegedly relaxed.
Shadow home secretary Yvette Cooper has written to Home Secretary Theresa May to ask how many terror suspects or illegal immigrants may have arrived.
It is alleged staff were told to relax identity checks on non-EU nationals.
Mrs May has ordered an independent inquiry into the allegations.
Brodie Clark, head of the UK border force and a board member of the UK Border Agency (UKBA), has already been suspended along with two others.
Mrs May is due to give a statement to the Commons on Monday about the situation.
In July, the Home Office said checks on EU nationals could be reduced in "limited circumstances", and their biometric passports checked "upon the discretion of a UKBA official" instead of automatically.
But it is now alleged Mr Clark told staff not to carry out checks on the biometric passports or fingerprints of thousands of people from outside the EU, heading for the UK, as they passed through the French port of Calais.
Mrs May's reaction to the allegations was "incredulity and fury", a source has told the BBC.
In her letter, Mrs Cooper said the public were "understandably appalled and shocked at the reports".
She said there were a number of questions the home secretary must now answer, first and foremost whether anyone posing a security threat had entered the UK.
"We need to know if someone on a watch list entered the UK during this time, and for the police and security services to take the necessary steps to protect the public," she wrote.
"Terrorist suspects have been seized at Calais in the past, as you will know.
"Abandoning the watch list checks raises serious questions about threat to our national security, and we need to know whether this allowed a suspect to enter our country who might otherwise have been caught."
Ms Cooper said the second step was to ensure the independent inquiry had a wide-enough remit to fully investigate "the actions of the Home Office, ministers and the effect of resource cuts on UKBA decision-making".
The third step, she said, was for the Home Office and UKBA to publish all documents and correspondence relating to the issue of passport checks.
Finally, she said, the inquiry must examine why it took four months for ministers to be made aware of reports that passport procedures were being ignored.
Labour and unions have claimed that staff shortages - due to cuts to the UKBA - are at the root of the problems.
Some 5,000 posts are due to go by 2015 as part of wider government cost-saving measures.
Lucy Moreton, from the Immigration Services Union, told the BBC: "Every passenger coming into the UK is supposed to be seen by an immigration officer, but there are instances where there are not enough staff and where, in fact, entire aircraft are missed, particularly flights that come into small, what we would call non-approved airports.
"We have a very large coastline... freight traffic coming into some of the large seaports can be missed.
"It's not supposed to happen, and often it doesn't happen on purpose, but it does happen."
The official investigation into the claims will be led by the Chief Inspector of the UK Border Agency, John Vine.