Add a Date field to the WordPress Customizer. Includes sanitize function for displaying date as '2016-10-25'.

WP Customizer - Date

$wp_customize->add_setting( 'date_setting_id', array(
  'capability' => 'edit_theme_options',
  'sanitize_callback' => 'themeslug_sanitize_date',
) );

$wp_customize->add_control( 'date_setting_id', array(
  'type' => 'date',
  'section' => 'custom_section', // Add a default or your own section
  'label' => __( 'Custom Date' ),
  'description' => __( 'This is a custom date control.' ),
  'input_attrs' => array(
    'placeholder' => __( 'mm/dd/yyyy' ),
  ),
) );

function themeslug_sanitize_date( $input ) {
  $date = new DateTime( $input );
  return $date->format('m-d-Y');
}