How to output comments number of a post per day?

There is no argument to restrict a comment query directly to a given date. You have to filter the query later:

/**
 * Get the number of comments for a post today.
 *
 * @param  int $post_id
 * @return int
 */
function t5_count_comments_today( $post_id = NULL )
{
    if ( NULL === $post_id )
        $post_id = get_the_ID();

    add_filter( 'comments_clauses', 't5_comments_today_sql' );

    // get_comments() returns a string even with 'count' set to TRUE.
    return (int) get_comments(
        array (
            'post_id' => $post_id,
            'count'   => TRUE
        )
    );
}

/**
 * Add date specific WHERE clause to comment query.
 *
 * Deactivates itself to avoid collisions with regular comment queries.
 *
 * @wp-hook comments_clauses
 * @param   array $clauses
 * @return  array
 */
function t5_comments_today_sql( $clauses )
{
    // avoid collisions
    remove_filter( current_filter(), __FUNCTION__ );

    global $wpdb;
    $clauses['where'] .= $wpdb->prepare( ' AND comment_date >= %s', date( 'Y-m-d 00:00:00' ) );
    return $clauses;
}

Simple usage

echo t5_count_comments_today();

Or, if you don’t want to show zero comments:

$comments_today = t5_count_comments_today();
if ( 0 < $comments_today )
    echo $comments_today;

Extended use case

This adds the number to the text of comments_popup_link().

is_admin() || add_filter( 'comments_number', 't5_comment_number_today', 10, 2 );

/**
 * Add formatted number to total comments number.
 *
 * @wp-hook comments_number
 * @param   string $output
 * @param   int    $number Might be a string, so we cast it to int.
 * @return  string
 */
function t5_comment_number_today( $output, $number )
{
    // no comments at all, no need to count today's comments
    if ( 0 === (int) $number )
        return $output;

    $count = t5_count_comments_today();
    $add   = 'no comments today'; // default text

    if ( 1 === $count )
        $add = 'one comment today';

    if ( 1 < $count )
        $add = "$count comments today";

    return "$output ($add)";
}

Side note: Always use get_comments() or the class WP_Comment_Query for which get_comments() is a wrapper, because that result will be cached, so you will never have to fetch the same value for an identical argument list twice.
Avoid non-API calls with raw SQL in these cases.