Back in the 1800’s people were more worried about what could get into their drinks rather than what could get on their furniture. Bear in mind, the luxury of really cold drinks is a fairly modern invention. Before refrigerators and easily available ice, drinks were merely kept cool, rather than cold. Thus condensation was not the main issue. The need of the hour was to keep bugs and dirt from getting into the pints.
Thus pubs started using squares of felt and other such materials to place on top of the pints. This was not the most hygienic of solutions because the often-soaked squares were used and reused ad nauseum without the benefit of proper washings in between.
It was a German printing company, Friedrich Horn that introduced the first cardboard beermats in 1880. Just two years later, the cardboard mat became out-dated, when Robert Sputh of Dresden, Germany introduced the wood pulp alternative. The onset of the industrial revolution allowed the wood pulp to be moulded and pressed by machinery, rather than the more cumbersome process of doing so by hand, thus ushering in the era of mass-production of beer mats.