Why is Dropbox erring on startup?
Google showed that it is the case with 10.3 while Dropbox only support 10.4 and upwards. 

Judging by the screenshot it is not your case so there is either a problem with zlib on your system or with permissions. Then either Disk Utility or booting from install disk and repair user permissions might help.

If nothing above try to remove Dropbox with a cleaner app and install anew.