The phrase ‘Curb Your Dog’ originated in the 1930′s in New York City. Citing (from barrypopik.com) the Chicago Daily Tribune, 4 December 1938, “Mostly About Dogs” by Bob Becker, pg. F10: “Curb Your Dog” Good Advice:
“In New York, truly a doggy city, an ordinance has been passed to make for a cleaner city and at the same time compel the indifferent dog owner to consider public welfare. The ordinance demands that dogs be curbed. There are signs everywhere with the request, “Curb your dog.” It means that owners cannot allow their pets to soil buildings, nor can a dog make a nuisance of himself on the grass of the parkway or on the sidewalk. As a result there are practically no complaints about the dogs soiling sidewalks or grassy places which the public uses.”
That is, ‘Please Curb your Dog’ meant ‘Don’t let your dog do its business on the sidewalk. Let your dog do it in the road’.
Urban Dictionary offers this definition:
How does this translate to you? It basically means, let your dog do his business in the "gutters" of the street and not in the middle of the sidewalk and clean up after your pet. Moral of the story? Don't be the disrespectful dog neighbor that let's your dog run wild and doesn't clean up after them.
You've also probably noticed that the Chicago Park District Doggie has gotten a face lift. He's lost his heavy chain leash in favor of a chic-er look. See the before (left) and current (right) signs posted around dog parks, parks, and public spaces.
For a more in depth analysis on what exactly "curb your dog" means, make sure to check out South Poop.