How to determine valid handle for given bottom up parser?
A handle is a production whose right-hand side can be reduced at a specific moment during a parse. (Or, as it is usually presented, it is the production which has been expanded at a specific point in a derivation for a sentence.) Although a handle is almost always defined as a production, it's common to talk about them as though they were just the right-hand side. That's clearly not correct because you cannot reduce a handle without knowing what it will reduce to, but in many contexts (including multiple-choice exams) it's possible to get away with this informality.

So I'd disagree that $B∗(id)$ is a handle, because it is not the right-hand side of any production. That doesn't really alter your question about whether $id$ is a handle.

A production (or its right-hand side) could be a handle for a grammar if there exists a derivation which involves that production. That is not necessarily the case for every production in a grammar, since grammars can include unreachable and useless productions. However, the grammar you present does not include any such productions. There exist derivations which use each of the productions, including $B\to id$ and $C\to (id)$. On that basis, I agree that both (A) and (C) are correct answers.