The first citation is in the New York Herald, but it appears that the idea reached New York from London.
(Oxford English Dictionary)
d. A person who takes dogs for walks as an occupation; also, one who rears puppies at home for a time before they are returned to kennels.
1930 C. FREDERICK et al. Foxhunting v. 61 The training of..'morale' begins from the day when the puppy returns to kennels from its walker.
20 June 1897, Boston Daily Globe, pg. 43:
NOVEL JOBS FOR WOMEN.
Some of Them are Dog Walkers, Some
Are Hotel "Porteresses," and Female
Cabbies are Said to be Coming.
It has been left to London to organize an institution that will give the new woman a decided new field.
The women's institute promises to be the most unique institution that ever was created for the gentler sex. Among the chief occupations which those who place themselves under its protecting wings will be trained in are those of dog walker and hotel porteress.
The first of these callings is as novel as it is surprising. There are no grounds attached to the homes of these dog owners, and therefore the only way for the pets to enjoy fresh air is to be walked up and down the sidewalk, after the fashion of the dogs belonging to the flat dwellers in New York.
Now the dog walker knows all about dogs. She calls at houses where she has learned canines are kept and presents a card, on which is printed in neat script, "Miss Dolly Periwinkle, Walker of Dogs."
-- New York Herald.
18 December 1897, Chicago Daily Tribune, pg. 16:
Dog-walking is the latest profession for women. The dog-walker sends her circulars to ladies, offering to exercise the canine pets at so much an hour.