Les Harris

The Pressbox plugin adds a new tab to the Media Upload screen that allows you to insert images and files from Dropbox into your posts.
Additionally, the plugin provides a new shortcode that lets you add links or display images anywhere you want on your blog.
This plugin is under heavy development so please feel free to send me feedback, bug reports, and feature requests!
Credits:
This plugin bears a great debt to the work of Joe Doleson’s Wp To Twitter plugin for his excellent example of OAuth integration with WordPress.

Pressbox

Integrates DropBox with your blog allowing you to post images and files directly from your DropBox. Requires PHP 5.2 and cURL.

acespritech

Are you looking for a way to create an upload form that automatically sends WordPress file uploads to your Dropbox? Using this Plugin you can easily integrate your Dropbox to your WordPress website. You can select maximum file size till your dropbox maximum upload size.
Features:- 
1 – Easy to connect form with Dropbox. 
2 – There is no file limit for upload. 
2 – Admin have short-code of form to display form in any page using it.
Shortcode:- [aspl_dropbox]

ASPL Dropbox File Upload

Another Best Plugin for Integrate Dropbox With Your Upload Form.

Pressbox vs ASPL Dropbox File Upload