This image of a threatening snowman is from a 1927 postcard. It was reproduced in a Smithsonian magazine feature about the history of the snowman in pop culture. From Smithsonian:
While no one knows for sure when exactly the snowman began smoking a pipe and drinking hard liquor, it may have started as early as 1890, based on a label from a bottle of whiskey from that year. An 1898 postcard shows a snowman carrying two bottles of champagne off to an office party. On holiday greeting cards from the 1900s through and on (up to the 1930s), the snowman often has a drink in one hand and a pipe in the other, mirroring our society's changes and America's fascination with smoking and drinking. This would eventually escalate to the snowman cavorting with women and offering drinks to minors. One could argue that these depictions were, in a way, humanizing, but seeing a tipsy snowman chasing a girl with a stick is disturbing at best.
By 1908, there was clear evidence of his partying ways were out of control. In the silent movie The Snowman by Wallace McCutcheon, a chain-smoking snowman is swigging whiskey and appears in the rest of the film sloshed, inspiring a flogging by the townspeople. This behavior would continue on film and media through magazines and postcards as a pickled, skirt-chasing, under-the-table lush. In other words, he had become a frozen W.C. Fields. By the '30s and '40s, there is no question, the two started to look alike, both wearing straw hats, putting on more weight and looking more round and sporting crimson noses. And both enjoyed prolific silent movie careers based on their reputations as charming drunks. It's hard to say if either had copied from the other but they were both enhanced by the other's notoriety. Ironically, W.C. Fields hated the holidays and passed away on Christmas Day, 1946.
UPDATE: In the comments, Bob Eckstein, the author of the Smithsonian article, points out that his book, The History of the Snowman, explains how snowmen developed their drinking problem. He says that his book is filled with many more fun images and deep insight into the secrets of snowmen. Bob also suggests we check out his Webzine, appropriately titled Today's Snowman.