One woman missed a flight to see her mum for the first time since lockdown despite queuing for three hours.
Donna Mayfield said the situation was "horrendous", while another traveller saw "customers and staff in tears".
Manchester Airport apologised and admitted passengers' experiences fell "below the standard we aim to provide".
Travellers at the airport have faced long queues for check-in and security over the past month, with some missing flights at the start of the Easter school holidays.
An airport spokesperson said: "Our whole industry is facing staff shortages and recruitment challenges at present, after the most damaging two years in its history.
"The removal of all travel restrictions after two years, coupled with the start of the summer travel season, has seen a rapid increase in passenger numbers, which is putting an enormous strain on our operation."
Ms Mayfield was due to fly out to see her 83-year-old mother in Spain on Saturday, after they could not meet in the past two years due to coronavirus restrictions.
"She's very, very upset, which is the only reason why I have booked on another flight today because I was just going to get refund on my flights.
"To be in a queue is bad enough anyway [but] you can see that people are getting more and more frustrated.
"I do sympathise with the staff, it's not like the staff weren't trying to do their best."
Other travellers told the BBC that the situation at the airport was "shambolic" and "shameful".
One man said he spent three hours in the security fast track lane without moving and missed his flight.
"It's not good for the travel industry or for the local Manchester economy… If I treated my customers like they did, I'd be out of business."
Another man, who spent seven hours at the airport before missing his flight, said he saw "sick on the floor with no one clearing it up" as people queued in a "boiling hot confined area".
Councillor Pat Karney recently blamed a "failure of management", saying the airport should have prepared for a "very quick" rise in travel demand.
A spokesperson for Manchester Airport, which is part-owned by the region's councils, said: "We are doing all we can to recruit the staff we need to meet this demand, but this is taking time due to the lengthy vetting and training processes involved."
The airport said it would also "continue to support" baggage handling agents, who are facing similar challenges.
Elsewhere, passengers faced delays at Heathrow Airport, which said the disruption was due to Covid documentation checks required by destination countries and higher numbers of passengers.
But travellers also reported problems due to staff shortages and e-gate passport checkpoints.