First, verify that the 404 is correct. You can do this by using a couple different HTTP status checkers (there are many free ones online) and seeing the response. (In some rare cases, Monitor Backlinks might say there is a 404 even though the page is definitely returning a 200 response. This is what you should do in that case.)
The 404s usually happen when you move a blog post, or change the URL structure of your site.
In that case, you should set up a permanent 301 redirect. It should be from the landing page with the 404 status to the corresponding new URL for the same page. If no such corresponding URL exists, you can also just 301 to another page on your site.
The 301 redirect will ensure that most of the link juice from the backlink is preserved.
Please note in some cases, the 404 might appear because you imported a link, and made a typo when importing the link. In that case, you should just edit the link within Monitor Backlinks and fix it.