Firstly the Dropbox sync is not a backup. It is a method used to share data between clients. If you delete something by mistake on one client that error will get propagated to all your other clients.
If you are recovering from a total loss of your system, i.e. it's a new computer, just install Joplin and connect it to your Dropbox account again. It is a new install and will download the notes. Same goes if you do a full re-install of Linux Mint.
However if after repairing your setup you still have a folder ~/.config/joplin-desktop left over from the previous install it may be best to rename it before installing Joplin. This is so any local and possibly corrupted data does not cause problems for Joplin or get sent to Dropbox by your repaired machine. If a new installation of Joplin finds that joplin-desktop folder it will try to use it and will not act as a new install but as if Joplin had been upgraded.