In general, you would use hook_form_alter() or hook_form_FORM_ID_alter() in a module to change all forms of a broad type. For all blocks, it looks like the form ID is always "block_form". So you can easily add elements to all block forms by implementing this function assuming your module is named "MYMODULE"
MYMODULE_form_block_form_alter(&$form, \Drupal\Core\Form\FormStateInterface $form_state, $form_id)
Since most blocks are extending BlockBase there is no way to change the class on the fly.
A really heavy-handed alternative to the form alter would be to use the decorator pattern and change all block plugins via the alter hook in discovery to use your class that adds some form elements and delegates most method calls back to an instance of the original class.