Contrary to popular belief most computer break-ins by external hackers don't happen because of great knowledge of operating system internals and network protocols. They happen because a hacker went digging through a company's garbage and found a piece of paper with a password written on it, or perhaps because they talked to a low-level bureaucrat on the phone, convinced this person they were from the local data processing department and that they needed him or her to change their password to "DEBUG."
This is sort of attack is called social engineering. Java applets introduce a new path for social engineering. For instance imagine an applet that pops up a dialog box that says, "You have lost your connection to the network. Please enter your username and password to reconnect." How many people would blindly enter their username and password without thinking? Now what if the box didn't really come from a lost network connection but from a hacker's applet? And instead of reconnecting to the network (a connection that was never lost in the first place) the username and password was sent over the Internet to the cracker? See the problem?