I use iOS with Keepass1. Not sure how you can do it with Keepass2 without converting the thing over and over.
Not sure what you mean by 'converting over and over' but MiniKeePass on iOS supports both 1.x and 2.x file formats. You do have to import the file from dropbox manually, however.
On iOS I use KeePass Touch. It syncs with Dropbox, and allows you to unlock the database with your fingerprint. At the time I searched, it was one of the only apps that fit these two requirements. Still works fine.