Is there any way to run hook_form_alter based on a form ’s class instead of it’s ID?
No - forms are only referred to by their ID (or 'base ID'), which is essentially the name of the function that defines them, or generally speaking, closely related to it.

So if you have

function MYMODULE_some_form($form, &amp;$form_state) {
  // ...
}


Then the ID of that form is MYMODULE_some_form. It's not related to the HTML ID attribute, in case that's what you're thinking, so there's no class equivalent.