Pass a parameter to a menu walker

As @toscho said, you can call the walker class with parameters as you did:

new BEMwalker( 'mobile' )

The constructor of BEMwalker will take the arguments (like any other function or method in PHP) so you can access the parameter(s) via $this:

class BEMwalker extends Walker_Nav_Menu {

    private $classes;

    public function __construct( $classes="" ) {
        $this->classes = $classes;
    }

    public function start_el( &$output, $item, $depth = 0, $args = array(), $id = 0 ) {
       $output .= sprintf( "<li class=\"%s\"><a href=\"%s\">%s</a></li>",
            $this->classes,
            $item->url,
            $item->title
        );
    }

}

Further reading: There’s also a GitHub repository called WordPress BEM Menu which might help you to implement a BEM-like syntax.