Are there good libraries in Beekman? Where?

Yes, the Beekman Library seems to be a nice place. The physical address says it's in Hopewell Junction, but don't be fooled, it's nowhere near the active village that most people would recognize as being HJ. This library is definitely in Beekman (the hamlet of Poughquag), near the intersection of Route 55 and County Route 9.

It's part of the local area's library system, which includes dozens of other libraries. This means your Mid-Hudson library card is good at any one of them and inter-library loans are at your disposal in case the Beekman Library doesn't have what you specifically need.