I had no idea that dog walking could be pinpointed to one person. But according to this New York Times obituary, the claim belongs to Jim Buck, who professionalized dog walking in New York City, and by extension in the United States, in the 1960s:
Starting in the early 1960s, Mr. Buck, the scion of a patrician Upper East Side family, rose each morning at dawn to walk passels of clients’ dogs, eventually presiding over a business in which he and two dozen assistants walked more than 150 dogs a day.
When he began that business, Jim Buck’s School for Dogs, it was the only one of its kind in New York. Today, the city has scores of professional dog walkers.
During the 40 years Mr. Buck ran his school, he was an eminently recognizable figure: an elegantly turned out, borzoi-thin man of 145 pounds, he commanded the leashes of a half-dozen or more dogs at a time — a good 500 pounds of dog in all — which fanned out before him like the spokes of a wheel.
Jim’s school is no longer in existence. But his legacy lives on: some of the city’s professional dog walkers are his former employees.