Unable to Upload the File into DropBox From Salesforce Using /Files_put API ?

First, you're using basic auth with your app key and secret, but this endpoint requires access to a specific Dropbox account, which is accomplished using an account-specific access token. You can find more information on using OAuth with the Dropbox API here:

https://www.dropbox.com/developers/reference/oauthguide

If you need to implement the OAuth 2 flow manually, this blog post may be helpful:

https://www.dropbox.com/developers/blog/45/using-oauth-20-with-the-core-api

The actual error you're getting indicates that the server didn't receive any data in the request's body. It looks like you're trying to set it using request.setBodyAsBlob, but it appears that didn't work, since the server didn't receive anything. You'll need to debug your usage of HttpRequest to determine why nothing is getting sent up.