final int nNames = 5; //just to make it easier to edit later
    String[] name = new String[nNames];
    String doB = new String[nNames];
    int counter = 0;
    do{
      name[counter] = JOptionPane.showInputDialog(" Enter Name ");
      doB[counter] = JOptionPane.showInputDialog(" Enter Date of Birth ");
    } while (counter < nNames);
    .
    .
    .
    outputArea.setText( "Name\tDate Of Birth" + "\n\n" + counter + "\t" ); //don't know what that should do, but anyway
    for (int i = 0; i < nNames; i++) { //that does exactly the same as your while loop above, but i is not visible after the loop, wich usually is what you want.
       outputArea.append(...); //name[i] and doB[i]
    }

*/ 
import javax.swing.*;
public class DateofBirth {
    
    
        
    public static void main(String[] args) {
 
        final int nNames = 5;
 
        
        String[] name = new String[nNames]; //name entered
        String[] dob = new String[nNames]; //DateofBirth entered
        
        int counter =0;
        
                  
            name[counter] = JOptionPane.showInputDialog(" Enter Name ");             
                    
            dob[counter] = JOptionPane.showInputDialog(" Enter Date of Birth ");
                      
                          
   
   JTextArea outputArea = new JTextArea();
        
   for (int i = 0; i <= nNames; i++)
 
            outputArea.append( "Name\tDate Of Birth" + "\n\n" + name + "\t" + dob );           
        
                              
        
        JOptionPane.showMessageDialog(null, outputArea,"Name & Date Of Birth",JOptionPane.INFORMATION_MESSAGE);
        
        System.exit(0);     
        
        
        
    }
    
}