How can I enqueue a style only when a particular widget is active?

You need to modify this a little bit.

class Organiser extends WP_Widget {
    ...

    function __construct() {
        add_action( 'wp_enqueue_scripts', array('Organiser', 'register_plugin_styles') );
        ...
    }

    public function widget( $args, $instance ) {
        wp_enqueue_style( 'organiserStyle' );
        include(dirname(__FILE__)."/organiserWidget.php");
    }

    static function register_plugin_styles() {
        wp_register_style( 'organiserStyle', plugins_url( 'organiser/css/organiserStyle.css' ) );

        // wp_register_style( 'organiserStyle', plugin_dir_url( __FILE__ ) . 'css/organiserStyle.css' );
        // wp_enqueue_style( 'organiserStyle', plugin_dir_url( __FILE__ ) . 'css/organiserStyle.css');
    }

    ...
}