1) You cant run that code as an application. Its an applet.
2) If you dont want to be able to copy at all do this:
   ta.setEditable(false);
   That makes it impossible to edit the text in the area but you can stille
copy with the mouse.
   ta.setEnabled(false);
   Make the field not respond to any user action.

Looks like bug 6300270 to me.

http://bugs.sun.com/bugdatabase/view_bug.do?bug_id=6300270

...

'Security'. *

Possibly.

* ..applets work in a restrictive sandbox that limits
what they can achieve if they are not signed.

An untrusted applet cannot do things like..
- open a connection to a foreign server
- access a variety of user or system properties considered private
- open files on the client's computer
- tap into the user's audio streams
- get information from, or put information on, the clipboard ..

..and that is the problem here.  An (unsigned?) applet
is not allowed to access the clipboard.

You mention both applications and applets but fail to
provide an SSCCE that can be run as both?!

<admonishing>Tut, tut..</admonishing>

You can confrim what I am saying by runnning *this*
(crudely hacked variant of the earlier) source, as either
an applet or application in your VM of choice.

The application will allow you to copy the text, whereas
the applet will silently ignore you.

<sscce>
import java.awt.*;
import java.awt.event.*;
import javax.swing.*;

public class DialogApplet
 extends JApplet
 implements ActionListener {

 JDialog d;

 public void init() {
   JButton b = new JButton("Show Dialog");
   b.addActionListener(this);
   this.getContentPane().add(b);
 }

 public void actionPerformed( ActionEvent ie ) {
   Frame f = Frame.getFrames()[0];
   JTextArea ta = new JTextArea("Are you able to copy this?");
   d = new JDialog(f, true);
   d.getContentPane().add(ta);
   d.pack();
   d.setLocationRelativeTo(this);
   d.setVisible(true);
 }

 public static void main(String[] args) {
     JFrame f = new JFrame( "Dialog Copy test" );
     DialogApplet applet = new DialogApplet();
     applet.init();
     f.getContentPane().add(applet);
     f.pack();
     f.setVisible(true);
 }
}
</sscce>

The solutions are to either
a) sign your applet so it is again trusted, and thereby
released from the restrictive security sandbox.
b) launch you applet using JWS, which might have a
different behaviour to copying in an unsigned applet.

Note that copy/paste in applets has never been that important
to me that I was bothered to test either approach - they might
work to restore copy/paste behaviour, they might not.

( And as a tip - *always* mention applets in the subject line,
if an applet is involved.  Where any problem involves an applet,
there is a >90% chance the problem is *due* to it being an applet. )

HTH

Andrew T.