It looks like that applets under Safari 7 are now sandboxed whatever the trusted certificate.
It breaks file chooser support.
A workaround is to enable unsafe mode for the web site (see screenshots) or switch to FireFox.
See also:
http://stackoverflow.com/questions/1979 ... -in-safari
http://www.engadget.com/2013/10/24/flas ... mavericks/