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.

You can tell if wood has been painted with latex paint by applying rubbing alcohol to a cotton swab then rubbing the cotton swab on the painted area. If the paint begins to peel of becomes tacky to the touch, it is latex. If not, more then likely the paint is oil-based.