I have never seen evidence of a scheduled through passenger train between Binghamton (or Wilkes-Barre) and Albany, but there were some close connections at Albany and Troy (yes, some A&S trains ran to Troy at one time).
Even in the early 1960's, you could ride 34 from Montreal to Albany and make a connection of less than 30 minutes to 208 for Binghamton. If 34 was in the station, they had to hold 208 for the first class mail from 34. There were no through cars, passenger, mail or otherwise, but the connection was made across the platform at Maiden Lane Station (Albany tracks 10 and 11). The North Albany Yard Engine set 34's rear up on the upper level for NYC to take to New York.
At least into the 1920's, the D&H had a comprehensive series of connections over the whole railroad, hinging on Trains 1 and 2 on the R&S. If you caught one of those connecting trains, you could go almost anywhere elso on the D&H before midnight, even Rouses Point, Rutland or North Creek to Wilkes-Barre or Binghamton.