There’s probably more than one approach to this. However, the way I do it on my site (the WP-Members support site rocketgeek.com runs on WP-Members) is to use the wpmem_register_form_buttons filter to change the register form button HTML to include the ga onClick event handler.
add_filter( 'wpmem_register_form_buttons', function($string,$tag){
if($tag=='new'){
$old = '<input id="button" name="submit"';
$new = "<input id=\"button\" name=\"submit\" onClick=\"ga('send', 'event', 'button', 'click', 'register');\"";
$string = str_replace($old,$new,$string);
}
return $s;
},10,2);
There are several other filters before (when the form is generated) and after (when the success dialog is served) that can be used for adding in ga tracking codes.