To 'put down' has meant 'kill' in a general sense since the 16th century. However, the first time it was used to refer specifically to the deliberate destruction of an animal was not until the end of the 19th century. Then, it referred to a dog that worried sheep having to be 'put down'.
It is also used in the following joke - A man takes his bulldog to the vet advising that the dog seems to be having trrouble seeing clearly. The vet picks up the dog and holds it in front of him peering into its eyes for several minutes. Finally the vet announces "Sorry, I'm going to have to put your dog down." Why?" asks the incredulous owner, "Because his eyesight is bad?" "No," says the vet. "Because he's really heavy!" I thenk yaou!