Sorry for delay replying! Yes, on some PCs the Java security warning happens with all Crossword Compiler applets run on Sun/Oracle Java.
Machines running upgraded and/or re-installed Windows XP SP3 seem the most prone to this error. Fresh Windows 7 machines seem ok.
The answer is here:
http://www.java.com/en/download/help/er ... edcode.xml
So you go to the Windows Control Panel, click Java, and on the Advanced tab click Security . . Mixed code, then select 'Enable - hide warning and run with protections'. Then all is well . . . . so far!