The word "mind" can have a few different meanings depending on the context, but in this case, the word "mind" is used as a verb. If someone asks "Do you mind?", it means they are inquiring if something bothers or inconveniences you. Essentially, they are asking if you have any objections or if something is okay with you.
Note: The provided meaning of the answer is for the latest New York Times crossword puzzle clue: "Do you ___?". Explanations are generated by AI and may be inaccurate.