It's a standard theorem that it's undecidable whether a given context-sensitive grammar generates the empty language or not.
From this, we can prove that it's undecidable whether a context-sensitive grammar is finite or infinite. Let $L$ be a context-sensitive language. By the standard closure properties for CSL's, if $L$ is context-sensitive, then so is $L^+$. Now if $L$ is empty, then $L^+$ is empty (and thus finite); but if $L$ is non-empty, then $L^+$ is infinite. Consequently, if it were decidable to check whether a CSL is finite or infinite, we could apply that decider to $L^+$ and learn whether $L$ was empty or not -- but the theorem mentioned above implies this is impossible.
Therefore, determining whether a given context-sensitive grammar generates a finite or infinite language is undecidable.
Footnote: You can find a proof of the standard theorem above in many places; e.g., these lecture notes from CS 373 at U Illinois. The result is also mentioned in passing on Wikipedia, which cites Hopcroft and Ullman -- a standard resource on formal languages. The same Wikipedia article also mentions some smaller classes of languages that are contained in CSL.