Do broken links on your site affect your SEO?

Yes, they do.

They lose you Google points, plus they increase your bounce rate because your visitors get frustrated and click away.

To check for broken links, either have a look in your Google Search Console, or use the Broken Link Checker plugin, but only do it once every six months or so, and when you’ve used it, uninstall it until the next time – it’s a very resource-heavy plugin so will slow your site down horribly if you leave it in place.