This is rather difficult to tell as modern water based paints are almost as durable as older oil based paints. If you see signs of peeling and the peelings look flexible and rather plastic like, you definitely have acrylic or water based paint. Older, cheaper water based paints were not durable and peeled easily off the surface. If the paint adheres well, the surface has a good sheen to it, you probably have oil based paint.
How can I tell if it is oil based or water based paint?