Send to Dropbox isn't the first email-attachment-to-Dropbox service we've seen, but it does have its advantages. While adding to your free file-syncing stash, it can also save original emails, unzip archives and organise your files by sender, date or other criteria.
Head to Send to Dropbox, authorise it with your Dropbox account, and you'll get a unique email address to send or forward attachments to — all well and good. After authorising though, you can set up options for including plain text or HTML email copies in the same folder as your files, as well as organising your folders. If you'd like ZIP-compressed files to be automatically un-packed, you can have that and keep the original ZIP archives in place as well.
Send to Dropbox is a free web-based service.