It should work. All it does is create a tailor-made URL, and then pass it to your default web browser, assuming your computer has one. Try copying the link directly from FishBase Client into your web browser. If that works, then the issue is with your operating system not accepting the URL, rather than FishBase Client itself. I've tested it on Mac OS X and Windows XP; other operating systems I can't vouch for.
One other thing: sometimes certain mirrors of FishBase won't allow a certain search -- that's why I have a row of buttons for the other mirrors. Try them instead. The ".us" and ".de" ones are usually the most reliable.
I didn't have any trouble. I have firewall issues at work, I imagine because of a firewall on the network wireless and firewalls on my programs. I can't find the hiccup so I just keep telling it "no, I don't want to deal with a runtime error."