A third of the dogs taken in by Battersea Dogs and Cats Home were put down last year, with the majority killed because they were too dangerous to re-home.
A total of 7,866 dogs were taken to the rescue centre in 2009, of which 2,815 (36%) were put down.
More than two-thirds (69%) of those destroyed - 1,931 - were healthy but judged too much of a risk to be offered for re-homing because of their temperament or behaviour.
Scott Craddock, the home's director of operations, told BBC One's Panorama that the animal centre was "mopping up" society's problems and added: "Enough is enough."
He said 3,600 Staffordshire bull terriers were taken in last year, nine times the 396 handed over in 1996, and described the rise as a "huge problem".
Bull breeds and bull breed crosses - many of them Staffordshire bull terriers - account for more than half of the home's longer term residents.
Mr Craddock said many of those dogs had to be kept in individual kennels which has a huge impact on the space available at the animal shelter, which is marking its 150th anniversary this year.
Taking in strays and unwanted animals cost the London-based charity £11 million in 2009.
Mr Craddock said: "Last year we reunited over 2,000 dogs to members of the public, those dogs which came in as strays. We re-homed 3,000 dogs, just over. But sadly over 2,800 dogs were put to sleep. Battersea is mopping up a lot of the problems that are happening outside of the home, big society problems, to such an extent that we feel it is time to say enough is enough."
Meanwhile Tim Wass, chief officer of the RSPCA inspectorate, told Panorama that 533 healthy dogs were killed last year. Unlike Battersea, which aims to accept every unwanted dog handed in, the RSPCA is now concentrating on helping the animals which most need its care.
