I've since discovered that this beautiful, charming set of houses was the least desirable place to live in mid-19th century New York City (they were built 1848–54). They "lacked all the vestiges of respectability—the houses were small, lacked the stoops so common among the brownstones of the day, and, most glaringly, were devoid of any prestigious street frontage." Its denizens were so poor that they couldn't afford a proper pint of ale and "had to resort to drinking a foul concoction—the nasty dregs that remained in their local barkeeps' beer barrel. Literally the 'bottom of the barrel.' Hence Grove Court's old moniker—'Mixed Ale Alley.' (It was also referred to as 'Pig Alley.')"
Apparently, Grove Court was built on an unusable little parcel of land that a shopkeeper named Samuel Cocks got the bright idea of creating "backhouses" on for workers—who would then patronize his store.
One interesting note is that Grove Court was the setting of one of O. Henry's most famous stories, "The Last Leaf."
This plaque, erected 1966 by the New York Community Trust, refers not to Grove Court but 4 through 10 Grove Street:
"These four houses, erected between 1827 and 1834, are typical of those built in New York in the early 19th century preceding the increased use of Greek Revival forms. Their simplicity of design is typically enhanced by enriched doorways. The street was deeded to the city as Cozine Street in 1809, opened as Columbia Street in 1811, renamed in 1813 in memory of Lieut. Burrows, naval hero of the War of 1812, and given its present name in 1829."