I use DropBox to sync the MBP and two iMacs. It has been the perfect solution for me but it is not idiot proof. I know I was warned by PAXTER1 but on two occasions it tried to open my DropBox BaseCamp Data with a previous version. BaseCamp warned me but both times I did not cancel so BaseCamp erased my data and started a new empty file. DropBox is very fast and before you can quite dropbox you have all computers synced to an empty data file.
Several ways to correct the problem after you get the BaseCamp updated on all synced computers.
1. DropBox keeps all file changes for 30 days but you can't restore the baseCamp data folder, you have to restore each file. I think the only files you need to restore is:
AllData.gdb
FolderData.gfi
2. DropBox is smart enough to make a backup copy in your DropBox folder. Just Delete the empty
AllData.gdb
FolderData.gfi
and replace with the backup file by removing the ( and everything between)
AllData (MBPro-7's conflicted copy 2011-06-03).gdb
FolderData (MBPro-7's conflicted copy 2011-06-03).gfi
3. Replace BaseCamp folder using TimeMachine.