Hi there,
if you use GenerateBlocks Pro then you can save the Container with the class in a Local Pattern.
Then you can add the pattern for each container.
But then you may also want to use a Global Style to sets on that Container , which would allow you to update the styles.
Otherwise, no you can’t add a class automatically .
But you could just target the default .gb-container class.
What styles are you wanting to manage from the customizer ?