Hi Mestika, which OS does your computer running?
If it is Mac OS, you can take a look at
SyncMate.
It cannot automatically send your SMS' backup to yuor Dropbox account, but it can export all of your messages into txt, csv, htm (and you can save this file into Dropbox folder). Please, note that this is just an export option, not backup (messages will be exported, but cannot be restored back to your device from this document).
One more note - this option is offered in Expert Edition only (in
SMS Manager plugin).
And what about SyncMate in general - this is a sync app for Mac and Android (and numerous other devices) which also offers you additional non-sync options like SMS management, possibility to mount Android device as a Mac disk, etc. SyncMate will sync contacts & calendars between Mac and Android for free; you can sync images, videos, music, folders, bookmarks in Expert Edition.
Details about Mac & Android sync with SyncMate
As the option you are interested in is offered in paid edition, we advise to test Free Edition prior purchasing Expert edition, in order to be sure your device is supported. Or you contact me directly at
alex.taylor_at_eltima.com and I will provide you with detailed description of SyncMate.