Installation 

This version supports TYPO3 11.5 LTS and 12.4 LTS with PHP 7.4 to 8.4 (as supported by the respective TYPO3 version).

Composer 

composer require svent/pageclasses
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Then update the database schema, e.g. in Admin Tools > Maintenance > Analyze Database Structure.

Without Composer 

Install the extension pageclasses from the TYPO3 Extension Repository (TER) in Admin Tools > Extensions. The database fields are created during the installation.

Include the TypoScript 

Add the static template Pagewide CSS Classes (pageclasses) to the main TypoScript template of your site: Web > Template, edit the whole template record, tab Includes, field Include static (from extensions).

That's all: the TypoScript of the extension sets page.bodyTagCObject (see setBodyTagCObject), so the classes are rendered without further configuration.