Extend boxes box, implement hook_boxes_plugins().

multilingual_box.inc

<?php

/**
 * Simple custom text box.
 */
class multilingual_box extends boxes_box {
  /**
   * Implementation of boxes_content::options_defaults().
   */
  public function options_defaults() {
    $defaults = array();
    $languages = language_list('enabled');
    foreach ($languages[1] as $language) {
      $defaults += array(
        $language->language . '-body' => '',
        $language->language . '-title' => '',
      );
    }
    return $defaults;
  }

  /**
   * Implementation of boxes_content::options_form().
   */
  public function options_form() {
    $form = array();
    $languages = language_list('enabled');
    foreach ($languages[1] as $language) { 
      $form[$language->language] = array(
        '#type' => 'fieldset',
        '#title' => $language->name,
      );
      $form[$language->language][$language->language . '-title'] = array(
        '#type' => 'textfield',
        '#title' => t('Box title'),
        '#default_value' => $this->options[$language->language . '-title'],
        '#description' => t('The title of the box as shown to the user.'),
      );
      $form[$language->language][$language->language . '-body'] = array(
        '#type' => 'textarea',
        '#title' => t('Box body'),
        '#default_value' => $this->options[$language->language . '-body'],
        '#rows' => 6,
        '#description' => t('The content of the block as shown to the user.'),
      );
    }
    return $form;
  }

  /**
   * Implementation of boxes_content::options_form().
   */
  public function render() {
    global $language;
    
    // Title. English is fallback.
    if (!empty($this->options[$language->language . '-title'])) {
      $title = check_plain($this->options[$language->language . '-title']);
    }
    elseif (!empty($this->options['en-title'])) {
      $title = check_plain($this->options['en-title']);
    }
    else {
      $title = NULL;
    }
    
    // Body. English is fallback.
    if (!empty($this->options[$language->language . '-body'])) {
      // Filter full HTML.
      $content = check_markup($this->options[$language->language . '-body'], 3, FALSE);
    }
    elseif (!empty($this->options['en-body'])) {
      $content = check_markup($this->options['en-body'], 3, FALSE);
    }
    else {
      $content = '';
    }
    
    return array(
      'delta' => $this->delta, // Crucial.
      'title' => $title,
      'subject' => $title,
      'content' => $content,
    );
  }
}

mymodule.module

<?php

/**
 * Implementation of hook_boxes_plugins().
 */
function mymodule_boxes_plugins() {
  $info = array();
  $info['multilingual_box'] = array(
    'title' => 'Multilingual box',
    'handler' => array(
      'class' => 'multilingual_box',
      'file' => 'multilingual_box.inc',
      'path' => drupal_get_path('module', 'mymodule') .'/boxes',
      'parent' => 'box',
    ),
  );
  return $info;
}