By default Divi only displays the logo image on your site, if you want to add a text below the logo, some sort of sub-heading, then you will need to use build a custom header using the Theme Builder feature.
First of all, make sure your Divi theme is up to date. At least, it should Divi 4.0 or a higher version.
All you need to do is go to the WP Dashboard / Divi/ Theme Builder page and create a new Global (or page-specific) Header:
Once you enter the Visual Builder mode and created the Header layout you like, simply add the text module for the site description part and enable the Dynamic Content feature under the Text Module settings and select the Tag Line in the drop-down list:
The Site Tagline text can be changed under the WP Dashboard / Settings / General / Tag Line option:
Fur more detailed info about Divi Custom Headers please refer to the following articles: