The free Broken Link Checker plugin from WPMU DEV is something you’ll pretty much want on every site. Broken links can occur at any time in old content because something they linked to is no longer available. This tool can scan on demand or automatically to let you know where you have broken links, and if you choose the automatic option, will send you email alerts when a problem develops.
There are probably other plugins that do the same thing, possibly just as well; this is the one I’m used to, and I know it does a good job. In the occasional situation where something looks broken and it’s really not (for instance when you’re linking to something that’s not visible to the scanner because it requires a login), you can tell it to disregard that problem, so it doesn’t keep nagging you about it.
As usual: if you know something better, please share in the comments!