How should I use wpdb class to submit a form in admin dashboard?

If you don’t want to use the Dashboard Widgets API you can use this code snippet that I have quickly concocted as a starting point:

function dashboard_daily_post_metabox() {
    add_meta_box( 'wt_id', 'Daily Post', 'dashboard_daily_post_process', 'dashboard', 'normal', 'high' );    
}
add_action( 'wp_dashboard_setup', 'dashboard_daily_post_metabox' );

function dashboard_daily_post_process() {
    global $wpdb;
?>
    <form method="post">
        <label>Age <input type="text" name="wt_age" value="<?php echo $wpdb->prefix; ?>" /></label>
        <input type="submit" class="button-primary" value="Save" />
    </form>
<?php
}

As you can see you have access to the global $wpdb variable and you can process your request as you wish.

You can quickly test this code by dumping it in your theme’s functions.php file.