Shadow home secretary Chris Philp said: “This is why it’s so wrong that French police stand by and do nothing while the illegal migrant boats embark – lives are being risked.”
Reform UK leader Nigel Farage said in a post on X: “We must put a stop to this. Our plan will save lives.”
Earlier this week, his party said if in government it would deploy the navy to ward off crossings.
Political discussions around the UK’s policies on illegal migration have resurfaced since tens of thousands of migrants swam into the Spanish exclave of Ceuta on Thursday.
The Conservatives have said that Labour has not delivered on its promises on migration, while the Lib Dems said the government should increase efforts to improve relationships with Europe and cut down the asylum backlog.
The number of migrants who have arrived in the UK on small boats since Burnham became prime minister a fortnight ago surpassed 2,000 on Saturday.
Last Wednesday was the busiest day for crossings so far this year, with 752 migrants arriving in the UK.
Most people attempt the crossing in the summer when conditions are better. It’s also become the most common way for people to be detected entering the UK illegally.
Home Office figures show the number of crossings so far this year are down about 43% compared to this time last year. The number peaked in 2022 and has been on a downward trend ever since.
During a visit to Dover on Monday, Burnham said progress had been made in curbing the number of people arriving in the UK on small boats.
The prime minister said “people want to see this issue addressed”.
He added that enforcement officers tackling people-smuggling gangs have been doubled on both sides of the Channel, but “a system that offers safe routes for people” needs to be built as well.
In April, the UK and France struck a £662m (773m euro) deal aiming to stop illegal migrants from crossing the Channel.
The three-year agreement included plans to send riot-trained police to French beaches to tackle violence and “hostile crowds”, as well as France deploying millions of pounds of drones and two helicopters as part of a system to intercept people smugglers and illegal migrants.
UK ministers have also said about £100m in funding could be redirected or withdrawn after a year if not enough journeys are stopped – but the government has not confirmed what targets the French would have to meet.