Passenger John Bailey told the BBC that the migrants tried to get access to the vessel by running up the main ramp.
But he said they were prevented from gaining entry when the crew raised the ramp and turned a fire hose on them.
The mayor of Calais said she would blockade the port unless Britain helped to control the migrants.
Natacha Bouchart said that her city was being “taken hostage” by more than 1,000 migrants attempting to cross the English Channel from France.
Officials said that two attempts were made to board the ferry. In the first about 85 migrants forced their way through a gate and climbed over fences, overpowering security staff reported BBC.
They were foiled when crew raised the ramp before they could get to the ferry.
A second attempt – made by about 150 migrants who also succeeded in gaining entrances to the port – was frustrated when police moved in to detain them.
“The migrants were escorted back down the ramp by the police and led out of the port, shouting at some people in the vehicle queue,” Mr Bailey told the BBC.
“The ship was delayed by about 45 minutes whilst a search was carried out.”
He said that on his way to the port he and fellow passengers “were shocked to see a huge crowd of migrants on the main access road who were being guarded by about 20 French police – some armed with sub-machine guns”.
“This was clearly no deterrent as they obviously ran past them into the port, which must have been a distance of about a mile,” he said.
The P&O ferry company, which had a vessel at Calais at the time, confirmed there had been a “huge intrusion” at the port and told the AFP news agency it had prompted it to close the doors to the ship.