It seems like many ancient civilizations did not take much precaution as to staying dry except for staying indoors. When they did go outside in the rain, it seems like none of their clothes were actually waterproof, and they just wore their normal outfits. Ancient soldiers and travelers didn't seem to mind much being soaking wet in their clothes. Perhaps this is due to a lack of watertight materials such as rubber/plastics/etc., or is this just a more modern idea? I can't recall any ancient civilizations where this isn't true and citizens, armies, etc. made a conscious effort to stay both warm, and dry.
Pretty much any Northern culture would be a very strong counterpoint to this. Norse and Sami were quite famous for their fur clothing, which kept them both dry and warm. Inuit in North America used sealskin to make winter garments only slightly inferior to modern materials. In fact, inuit clothing was used for arctic exploration until quite recently in the 20th century. Other groups would have used different waterproof materials like oiled leather or plants for water protection. Palm leaves are still used for this purpose in many areas today.
It makes sense that historic peoples did this, as human biology hasn't changed. People in the past were just as susceptible to hypothermia and frostbite as modern humans, both of which are risks associated with weather. Dampness in tropical environments also carries parasite and fungal risks.
A universal way of staying dry through many cultures is wool. It is surprisingly water repellant and can take in a lot of water before it feels damp. The ubiquitous wool cloak you find in most of history is popular because it does so well at keeping the wearer dry. Oiled and waxed fabrics also go back a while, but when something is "waterproof" it's also airproof, and just like that plastic poncho will feel sweaty and damp inside.