I recall watching trains on this track in the late 60's and maybe early 70's, but it was later shut down. They also removed the trestle over the road.

Does anyone know when they stopped using those tracks?

Also, what sort of rail traffic is there in Keene now?

I don't know when the trains stopped running, but a recent map (MapLink's Railroads of the Continental US-1996) shows no active rail lines in Keene anymore. The nearest would be the New England Central (ex-Central Vermont) along the Connecticut River.
But a look at MapQuest still shows a rail line through Keene, but this could be out of date info.