Chiedi ad alka di unire i due thread.

Comunque, non capisco perché tu estanda MenuApplet per creare la classe che gestisca il thread...
codice:
import java.io.*;
import java.util.*;
import javax.swing.*;
public class MenuApplet extends JApplet {      
  public void init() {         
    Thread_ShowStatus m=new Thread_ShowStatus(this);
    Thread t=new Thread(m);
    t.start();
  }  
}

class Thread_ShowStatus implements Runnable {
  
  private MenuApplet ma;
  public Thread_ShowStatus (MenuApplet ma) {
    this.ma = ma;
  }
 
  public void run() {
    while(true) {
      try {
        ma.showStatus("Click on submit button howewer.Click on submit button." );
      }
      catch(Exception e){}  
  }
}   
}