Boss Doug Bannister says the port has been "badly let down" by French border controls.
He said officials had been "insufficiently resourced" in France and working slower than normal, leading to delays of up to five hours.
The Kent port has been hit by six-hour queues, with tourists urged to consider staying away.
It is one of the busiest periods for foreign travel from the UK as most schools in England and Wales break up for summer this week.
Gareth Fuller
Passengers embarking on cross-Channel sailings from Dover must pass through French border checks before they can board a ferry.
The port said in a statement that it has increased the number of border control booths by 50 percent and shared traffic volume forecasts “in granular detail” with the French authorities.
It went on: “Regrettably, the PAF (police aux frontieres) resource has been insufficient and has fallen far short of what is required to ensure a smooth first weekend of the peak summer getaway period.”
Delays at Dover are causing tourist and freight traffic to be stuck on gridlocked roads in the area.
Gareth Fuller
P&O Ferries told passengers: “Please be aware that there is heavy traffic at border control in the port of Dover.
“If you are booked to travel today please allow at least six hours to clear all security checks.”
It comes as fuel price protests are set to cause further disruption.
Avon and Somerset Police warned motorists that “slow-moving roadblocks” are planned on parts of the M4, M5, M32 and A38 on Friday morning.
An image posted on Facebook group Fuel Price Stand Against Tax suggests demonstrations will be held “nationwide”, including in Birmingham, Cardiff, Liverpool, London and Manchester.
The RAC said an estimated 18.8 million leisure trips are planned in the UK between Friday and Monday.
That is the most since the company began tracking summer getaway numbers in 2014.
Superintendent Tony Blatchford, of Avon and Somerset Police, urged drivers to consider “alternative travel plans” due to the pump price protests.
He said: “Our protest liaison team has been engaging with the organiser so we can inform the public of the likely disruption and help to minimise it.
“Nevertheless, drivers can expect journey times will likely be longer than normal, especially on motorways, which often tend to be at their busiest at this time of year.
“We advise motorists to consider any alternative travel plans available and ensure they are suitably prepared in case they are delayed.”
Fuel price protests on July 4 led to 12 people being arrested on the M4.
