How to add CSS classes to buttons of Drupal Commerce checkout forms?
According that you have default views_form_commerce_cart_form_default form for your cart, this should do the job. Put the following lines in a custom module:

function MYMODULE_form_views_form_commerce_cart_form_default_alter(&amp;$form, &amp;$form_state, $form_id) {
  $form['actions']['checkout']['#attributes']['class'] = array('btn-success');
}


If not ok, check the ID of your form with 

function MYMODULE_form_alter(&amp;$form, &amp;$form_state, $form_id) {
  drupal_set_message("Form ID is : " . $form_id); 
}


and apply the same above snippet with function MYMODULE_form_FORM_ID_alter.

EDIT : to apply btn-success to Continue buttons of the cart, checkout, review pages: 

function MYMODULE_form_alter(&amp;$form, &amp;$form_state, $form_id) {

  switch ($form_id) {

    case 'views_form_commerce_cart_form_default':
      $form['actions']['checkout']['#attributes']['class'] = array('btn-success');
      break;

    case 'commerce_checkout_form_checkout':
    case 'commerce_checkout_form_review':
      $form['buttons']['continue']['#attributes']['class'] = array('btn-success');
      break;   
  }
}  


To do it yourself next time : in your hook form alter() module, check the form_id, dsm($form); and browse the dsm tree to find your needle...