Google uses headers to understand the relevancy of your page for a user’s search query, so they certainly affect SEO.
Google hints in its Search Engine Optimization (SEO) Starter Guide that it’s more about how large the text is relative to other content, though within most CMS platforms, it makes sense just to use header tags.
As John Mueller from Google says, “Headings on a page are great for SEO & accessibility, but they’re not going to make or break your site’s rankings… H1 elements are a great way to give more structure to a page so that users and search engines can understand which parts of a page are kind of under different headings.”
Though rare today, sometimes the largest text on-page is something lacking keywords like BLOG or ARTICLES. This text may be within a banner describing the whole section, not the specific content itself. Avoid this!
The real opportunity in using headers today is to increase the likelihood for more search engine real estate via rich or featured snippets. You should apply headers by writing through an outline format. It gives Google a clear structure to understand that the paragraph, list, or table below the header is answering a question related to that topic.