On a recent question, a user decided it was a bad question and downvoted all answers with the comment "-1 closevote, not answer!". I realize that the community doesn't want to reward help vampires, but is this the appropriate response?
Don't mass-downvote answers. It's just bad form.
If you hover over the downvote arrow for an answer, the tooltip says "This answer is not useful." That should be your guidance for downvoting answers.
We don't have a "policy" that users should vote to close instead of answering an off-topic question, though that is probably a good practice to follow. We certainly don't have a standing policy that users downvote all answers to off-topic questions; that is not the purpose of voting.
That said, it's perfectly within the right of a user to downvote any answer for any reason, just as it is perfectly within the right of a user to post an answer to an off-topic question, so long as that question is still open.
Down-voting most of the answers given for a question (or all the questions) just on the basis somebody thinks the question should be closed doesn't seem fair, to me. Down-voting should be reserved for answers that are not useful.
Leaving out the specific case, what would happen when a user thinks a question should be closed, but other users think it should not be closed, and they don't vote to close the question, which doesn't reach the necessary number of closing votes? What happens if the question in the current form needs to closed, it gets closed, the OP edits it, and the question is re-opened? In this case, the question still needs answers, and the existing ones could still suit the question, or need to be edited to suit the question as it has been edited.
In both these cases, who answered the question got a down-vote that cannot be removed if they don't edit their own answer, and the question is still open. Do you think it is fair? I think it is not.
Looking at the specific case, the user down-voted also the answer given from a user with a reputation lower than 3 thousand, who cannot vote to close questions, and cannot see when a question is being voted to be closed.
If you look at the example you mentioned, it seems pretty counter-productive to mass-downvote answers: A question that would have been closed almost immediately without much ado now has a massive 10 downvotes (I feel sorry for the OP...), downvotes on the answers that lead to sympathy upvotes (I think...) that lead to more downvotes that lead to more upvotes.
Kind of the opposite effect of what you want for a question like that in my opinion.