Not all downvoted answers should be deleted.
Downvoted answers (especially with comments) can add value. They basically say "here is one approach to solving your problem. It's not a good idea; don't do it."
One example is this answer about working for free. The community generally agrees that it's a bad idea (shown through the downvotes), and there is a comment explaining why. Clearly, the idea occurred to at least one person as a solution to the problem, so it's useful to keep around something refuting that idea.
Rather than go for deletion of such answers, I'd recommend the first action be to try to add comments to explain why it's a bad idea. If that's not a viable option, then deletion may be appropriate.
In the FAQ on Comments on meta.SO comments are clearly intended to be temporary (emphasis mine):
Comments are disposable: unlike posts, there's no revision history, and they can be deleted without warning by their authors, by moderators, and in response to flags.
When should comments be deleted?
Comments are temporary "Post-It" notes left on a question or answer. You should not expect them to be around forever: Once a clarification has been made, an edit added to the post to include new information, or the issue in the comment is otherwise resolved, it is subject to deletion. In reality, many obsolete or chatty comments remain untouched due to the high volume of comments posted, but this does not mean that they can't or shouldn't be deleted in the future.
Personally I share gnat's take on things:
I believe that it would be better to have an (upvoted) answer that lays out the idea in the impartial tone and follows up with a compelling explanation for why it would be bad. If there was such an answer, I wouldn't mind deletion of the one that gives a pretty shaky "proof by comment and downvotes"
So the question I want to place to the community is:
Are 'Whatever you do, don't...' Answers Appropriate?
Taking the answer used as an example by David, would it be appropriate to make an answer saying something like:
While it may seem tempting, do not offer to work for free. Working for free may get your foot in the door, but it will cause the employer to value your work at nothing because you're providing it for free. This will hurt your chances to negotiate with the company for a full-time position
This does not answer the question, but it does provide valuable information that is currently hidden in a -7 downvoted answer, and a comment clarifying that the answer is a bad idea. Personally I think that having such an answer would be better than having the downvoted answer as-is, but I wanted to run it by the community first.