That plugin does not create a
<form> tag itself. It uses
wp_login_form to do that, and that function does not provide a way to add a class to the
<form> tag. You can see that by looking at the
widget method, which is what generates the front end content.
I don't see a way to do what you want to do, but there is an action before the form--
do_action( 'sidebar_login_widget_start' );-- and one after--
do_action( 'sidebar_login_widget_end' );-- that would let you wrap the whole form in some other markup.
However, I very much doubt that you need to add a class to that tag. I would bet that you can do anything you need by writing CSS rules using the existing markup.