About 1,000 people are staying in makeshift camps along the French coast, with many intending to try to cross to Britain.
A survey of migrants in Calais revealed that more than half (55 per cent) think they have a better chance of getting asylum since December 31.
Only 18 per cent thinking they had a worse chance, according to the same study.
The poll also revealed widespread fears, with a majority saying they felt less safe in 2021.
Immigration minister Chris Philp said “the claim that getting asylum is any easier now is categorically untrue”.
Since the start of 2021 six people have been sentenced for facilitating illegal journeys to the UK, the Home Office said.
Last year thousands of people tried to reach Britain from France, often making dangerous trips in unseaworthy dinghies or squeezing into lorries – with sometimes deadly consequences.
With Brexit, the Government vowed to end freedom of movement and “take back control” of the UK’s borders.
However, a majority of migrants in Calais quizzed earlier this month believe they now have a better chance of getting asylum in the UK than before it left the single market on December 31.
Those in Dunkirk were less confident, with 37 per cent saying they felt they had a better chance and 25 per cent a worse chance.
Across both locations nearly 80 per cent of respondents said Brexit has made physically reaching the UK harder.
Humanitarian charity Care4Calais conducted the poll of 139 migrants living in Calais and Dunkirk together with Dutch campaign group Stop Wapenhandel (Stop Arms Trade).
Upon leaving the EU, the UK also left behind the Dublin regulations, which allows member states to return asylum seekers to the EU country they first entered.
What is more, Care4Calais suggested that longer lines of lorries in Calais waiting to go to the UK will provide many opportunities for people to try to climb aboard, risking their lives as they do so.
In 2020 more than 8,400 people crossed the English Channel aboard small boats, data showed.