As of January 2014, Dropbox does not offer a default service for saving emails. To save email messages to your Dropbox account you will need to use a third-party tool. Most services save attachments from emails without including the body of the message, but some services allow you to forward an email directly from your smartphone to your Dropbox account, where the message is saved as a text file.
Third Party Services
One service for saving your emails to Dropbox is Send to Dropbox. It connects to your Dropbox account and gives you a unique email address for sending files to your account. In the Send to Dropbox options, select "Include plain text email body (if available)." You can opt for HTML if you prefer. Then forward the email to your Send to Dropbox address. Another option is to use the If This Then That service to send specific tagged emails to Dropbox; the service offers a specific recipe for forwarding emails to Dropbox. (See links in Resources.)
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