Google's John Mueller has an awesome line in a response to a Google Webmaster Help thread question about broken links.
John wrote, "The web changes, sometimes old links break. Googlebot isn't going to lose sleep over broken links."
John explained that GoogleBot can handle broken links but many of your users cannot, so make sure to fix them more for usability issues versus SEO issues. John wrote:
If you find things like this, I'd fix it primarily for your users, so that they're able to use your site completely. I wouldn't treat this as something that you'd need to do for SEO purposes on your site, it's really more like other regular maintenance that you might do for your users.
That being said, if GoogleBot only hits broken links on your site and your internal navigation is all broken, it may have a problem indexing and ranking your web site.
Forum discussion at Google Webmaster Help.
This post was pre-written and scheduled to be posted today. Have a happy, healthy and sweet new year!
Image credit to BigStockPhoto for broken links.