The biggest problem with this badge is that this is a really boring use of time that doesn't really improve SO that much and doesn't really lead to our users undergoing much growth as software developers. Creating a badge to incentivize something is not a magic way to produce user-hours. If it increases the number of hours our users put into the site, they should be quality hours so users enjoy using SO and get something out of it, neither of which will really be the case here. And if it detracts from other tasks, like answering questions or editing answers, it should be a better use of time, which I think is not the case with fixing broken links.
Our editors' time is better spent maintaining active posts than goose-chasing down broken links to obscure answers that are never read. It's much like maintaining a legacy code base: you don't really want to spend ROI-less hours doing "cleanup."
Furthermore, bear in mind that old answers are to some extent self-maintaining. An answer with a dead link that is generating many views will eventually be flagged as having a dead link or edited, so there is a natural cap on how much attention an old answer can get while in a (conspicuously) invalid state.
Now that all being said, if we do decide this is a problem we want to address, a review queue is a better approach than a badge. Many broken links have bad HTTP requests that an offline task can scrape and queue up for review. This is far preferable to users coming up with hacky ways to goosechase broken links; furthermore, a queue can order by, say, question views and better prioritize edit time. Per Caleb's comment I think this ought to contribute to copy editor, because I still don't think this is a more worthwhile task than other forms of editing.