Problem of determining whether a context sensitive language is context free is undecidable.
How to prove it
This can be done in a similar way to the proof that it is undecidable whether the language of a context-free grammar is regular. A general statement of such undecidability properties is Greibach's theorem observed by Sheila Greibach in 1968.
Let $C$ be a family of languages effectively closed under union and concatenation with regular sets, and for which equality to $\Sigma^*$ is undecidable (for sufficiently large $\Sigma$). Let $P$ be a proper subset of $C$, which contains all regular languages, and is closed under single letter quotient. Then membership of $P$ is undecidable for $C$.
So, we have to check whether the context-sensitive and context-free languages meet the criteria of Greibach for $C$ and $P$.