Is it the same URL you are accessing when you do get the dialog in developer?
If yes, it might happen because the client is running on an older Java version which doesn’t have stuff up-to-date. Try to run it using a newer Java. (you can for example manually run it by downloading the JNLP file instead of running it and using ‘javaws’ executable from your java install/bin dir - or just change the Java version on your system)
If not, then the URL that is the problem - can you access it nicely in the browser? Does it show any problems in the security info of the page (usually a lock symbol in browsers) or does it say it’s a valid encrypted connection?