Is your walker.php file included or required in functions.php? Did you instantiate your class and pass the object as an argument in the theme’s call of wp_nav_menu()?

Even if you have no use for the wp_nav_menu() arguments, for full compatibility, your method should collect the arguments and specify defaults anyway.
public function start_lvl( &$output, $depth = 0, $args = array() ) {

Then sometime down the road you can easily plug your class into any theme without errors or conflicts.