"The very lowest class of drinking-places are cellar groggeries called 'bucket-shops'; beer and spirits are sold in jugs, buckets, and bottles as well as over the bar, and in many of the shops the remains of drinks, together with the washings of the bar and the rinsings of the cloths with which it is wiped, are thrown into a tub and sold for two and three cents a quart. This liquid is known by fancy names, such as 'dog's-nose', 'all-sorts', 'swipes', and other terms."
--Darkness and daylight by Helen Campbell, Thomas W. Knox, and Thomas Byrnes, 1896.
Dog's Nose is actually two different drinks; it has been used since at least 1812 to refer to unwise mixed drinks, as described by the passage above. But when Dickens' Pickwick Papers was published in 1836, he described a Dog's Nose as "compounded of warm porter, moist sugar, gin, and nutmeg"; not the most respectable drink, but not a sin against alcohol either. Since that time, in England, the recipe seems to have stabilized as a common term for any mixture of beer or ale and gin (or sometimes, to mix it up, rum); this is probably a rather straightforward result of the overwhelming popularity of gin and beer as the unofficial national drinks of Britain. Charles Dickens seems to have enjoyed this drink himself, along with the same mixture with the addition of ginger, a concoction known as Purl.
As evidenced by the quote above, in America a Dog's Nose remained a disreputable drink for at least another 80 years. It eventually sunk out of fashion, and now the term is only used to refer to the mixed drink in the 'British' recipe. These days a Dog's Nose is usually given as one part gin to 12 parts beer, although sometimes the ratio is 1:10 or greater. A dusting of nutmeg is still recommended if you use stout. There is a false etymology that the gin-and-stout recipe gained the name Dog's Nose because it is wet and black, but this is likely not true. It is much more likely that the original spirit was more along the lines of 'a drink fit only for a dog'.