To start, paper bags aren't associated specifically with alcohol drinks, they are associated with the consumption of alcohol in public. The use of the brown paper bag covering a bottle of alcohol is meant to hide the alcohol while the imbiber is in public, as opposed to being in a tavern or the privacy of their own home. The term associated with concealing one's alcoholic beverage in a brown bag is called brown bagging. The potential for the mass production of paper bags began with the invention of the paper bag machine patented in 1852. The link above also mentions the invention of Margret Knight's design of an apparatus that affixes square bottoms to paper bags in 1870, giving them the style we are still familiar with today. So it's safe to assume that brown bagging did not predate this time.

Here we have the concurrent materialization of a technology and social reforms. A pamphlet by Benjamin Rush, a Philadelphia physician, was published in 1784 and introduced the negative effects alcohol had on society. The 19th century witnessed several Temperance Movements that would change the way American society viewed alcohol, consequently affecting its consumption. An illustration of The Opposer of Temperance, who is against Maine's Liquor Law, is depicted with a bottle of liquor in his hand, clearly without a paper bag. So even though the idea of alcohol as a social taboo is gaining momentum, it has yet to become so intolerable that the use of it in public warrants concealment. Also, as mentioned above, paper bags probably aren't in wide use at this time (1851).

The public consumption of alcohol was not the primary focus of temperance leagues, however. Their main concern was creating a completely dry society. The Anti-Saloon League, in cooperation with the Woman's Christian Temperance Union played major roles in the passing of the Eighteenth Amendment and the onset of prohibition.

With the consideration of temperance being against alcohol, and not just the public consumption of alcohol, I posit that the introduction of brown bagging came in the wake of prohibition and preceded public intoxication and open container laws, which vary by state. Merriam-Webster proposes that the first use of this term was in 1959. Not to say it wasn't prevalent before the term was coined.

Harper Lee refers to brown bagging in his novel, To Kill A Mockingbird(1960). She doesn't use the term, but uses the act of carrying alcohol in a brown bag in public to suggest that his character Radley is a drunk. Since the book is set in 1936, it is possible that brown bagging occurred at that time. Lee would have been a child in the 1930's but still old enough to witness public displays of drunkenness. Literature, whether fictional or not, is a great tool in discerning social practices and opinion.

For more information, here is an academic site that boasts several primary sources on the American Temperance Movement.

Edit: typo