It could potentially be a pretty good interview question to weed out those who have spent the last 6 months memorizing algorithms.
Realistically, anyone with programming experience would immediately question the premise of being asked to implement a binary search on a whiteboard because realistically the only way anyone will succeed is by having memorized an implementation.
I don't think this is an especially great question, but I worked at one company that asked this question and most people who got it right did not seem to have an implementation memorized. I say this because they got it right only after writing a buggy version and fixing some bugs.
Another company was in the habit of asking a different question that gets to the heart of the matter of "Can you think with invariants?" but is not nearly as tricky as binary search. That seemed better to me.
It is a very useful question if you do not simply check whether the result is correct, but instead use it as a way to evaluate the candidate's problem-solving skills.
Realistically, anyone with programming experience would immediately question the premise of being asked to implement a binary search on a whiteboard because realistically the only way anyone will succeed is by having memorized an implementation.