The original meaning of the word blanket was a rough undyed woollen fabric used in a variety of ways. Gradually this became used primarily for a bed-covering and in the early fourteenth century the word gained its modern meaning of a large rectangular piece of woollen or similar material (often with bound edges) used as a bed covering or other covering for warmth especially a large, rectangular piece of fabric, often with bound edges. To blanket then came to mean to cover with something and by extension the noun came to mean any extended covering or layer such as a blanket of snow.
In case of fire attempting to smother with a wetted blanket was (and remains) as effective as most things that come readily to hand. The term wet blanket therefore acquired its modern meaning of a person who smothered the enthusiasm of others by his mood or behaviour.
Meanwhile, to be born on the wrong side of the blanket had become an oblique way of saying someone was born illegitimately. Illegitimacy has always been a special case in heraldry – indeed, it has been a special case in Western Christianity, the group of societies in which heraldry first arose. When someone entitled to bear arms has acknowledged paternity of his 'natural children' or those “born on the wrong side of the blanket”, and has made the necessary arrangements with the heraldic authority, his illegitimate offspring would receive a grant of part of his arms ‘suitably differenced’. This was usually thought to be by having a bar or baton across them so their arms included a “bar sinister” - a broad diagonal stripe from top right to bottom left. In Victorian times this was held to be a terrible social disability. People from armigerous families who were of bastard descent would often forget or deny their connections in order not to be obliged to bear arms that included the said “bar sinister”.
This drawing is by John Pottinger (1919–1986), a Scottish officer of arms, artist, illustrator and author of 'Simple Heraldry, Cheerfully Illustrated'. He makes use of the expression “wrong side of the blanket” to produce an illustration of Campbell, Duke of Argyll, with two legitimate sons (both doing the Highland fling!), one with a label, the other with a plain blue border. Below the blanket is their illegitimate half-brother William, with a broken border and a “bar sinister” (and a filthy look to boot!).