When did the fashion of dressing children as little sailors start ?

Question

and where?

Answer 1

Actually, small boys were sailors - a powder monkey was a boy between 4 and 7 yrs of age who would fetch gunpowder for the cannons on war vessels in the 17th & 18th centuries.

If you are asking about the fashion of dressing upper class children in sailors outfits, it was fashionable during the regency and victorian period in England.  It spread from there to other places like the US.

Specialized children's clothing (other than infant wear) really wasn't developed until the late 1700s.

Answer 2

Given that sailors only started wearing uniforms post Napoleon, it is odd that it created such a fashion.

Wikipedia says

"In 1846, the four-year-old Albert Edward, Prince of Wales was given a scaled-down version of the uniform worn by ratings on the Royal Yacht. He wore his miniature sailor suit during a cruise off the Channel Islands that September, delighting his mother and the public alike. Popular engravings, including the famous portrait done by Winterhalter, spread the idea, and by the 1870s, the sailor suit had become normal dress for both boys and girls all over the world."