And it wasn't PETA that did them in:
Fur was falling from favour well before the activism of the 1980s. In the late 1950s the price of mink fell dramatically. The cost of manufacturing a mink coat now exceeded that of the raw materials and there were many in the trade who felt that the luxury status of fur was becoming a thing of the past. Demand began to fall. The widespread adoption of central heating no doubt played some part: in bitter cold, nothing keeps you warm like natural fur. But the truth was that the fur coat, once the epitome of glamour and luxury, acquired unfashionable connotations from the 1960s. It signified an older, less trendy and more dependent kind of femininity.
(Hat tip: Virginia Postrel)