Theme Customizer for only author.php (per user baisi)

First, you will need to generate customizer settings depending on the current user. So there are two conditions to be met: the setting must only be accessible to the current user and this setting must be unique to him.

// Only do this for contributors and up
if (current_user_can('edit_posts')) {
   // retrieve author id
   global $current_user;
   get_currentuserinfo();
   $author_id = $current_user->ID
   // generate customizer setting for this author only
   $wp_customize->add_setting( 'header_color_' . $author_id , array(
     'default' => '#000',
     'sanitize_callback' => 'sanitize_hex_color',
     'capability' => 'edit_posts',
     ));

In (the heading of) your author.php you will need to get the id of the author of the current page to retrieve the relevant mod.

$author_id = get_the_author_meta('ID');
get_mod('header_color_' . $author_id);