I actually don't believe I've come across any museums in Dover Plains. There are museums and art galleries in some nearby towns, however. For example, there are some of both in Kent, on the other side of the Connecticut border from DP, but I also know from first-hand experience that there's no direct route between the two and it'll be a bit of a drive (over 20 minutes). I also know there is something in the way of art out in Millbrook, which might also take about 15-20 minutes to reach.
I have found some individual artists in Dover Plains, but I'm uncertain if there are any public galleries in town that are affiliated with them.