Convert a Graphics2D to an Image or BufferedImage

Have them draw directly in a BufferedImage by way of it’s Graphics2D object which you can get via getGraphics(). Then use ImageIO.write(...) to output the image to whatever file type you desire (and that’s supported). The ImageIO API should help you with this: ImageIO API.

The other issue you’ll have is where are they supposed to save the image once it has been drawn? On their own computer? If so and this is an applet program, then the applet will need to be “signed” in order for it to have the permission to write to disk. If you’re unsure on this, check out Google, this article, or you may wish to write a new question for this issue alone.

Edit 1: code example
For example:

import java.awt.*;
import java.awt.event.*;
import java.awt.image.BufferedImage;
import java.io.File;
import java.io.IOException;
import java.util.ArrayList;
import java.util.List;
import java.util.Random;

import javax.imageio.ImageIO;
import javax.swing.*;
import javax.swing.filechooser.FileNameExtensionFilter;

@SuppressWarnings("serial")
public class DrawAndSaveImage extends JPanel {
   private static final int BI_WIDTH = 600;
   private static final int BI_HEIGHT = BI_WIDTH;
   private static final Color LABEL_DRAW_COLOR = new Color(180, 180, 255);
   private static final Stroke LABEL_DRAW_STROKE = new BasicStroke(1);
   private static final Stroke BIMAGE_DRAW_STROKE = new BasicStroke(4);
   private static final int COLOR_DIV = 5;
   private BufferedImage bImage = new BufferedImage(BI_WIDTH, BI_HEIGHT,
            BufferedImage.TYPE_INT_RGB);
   private List<Point> pointList = new ArrayList<Point>();
   private JLabel imageLabel;
   private List<Color> colorList = new ArrayList<Color>();
   private Random random = new Random();

   public DrawAndSaveImage() {
      Graphics2D g2d = bImage.createGraphics();
      g2d.setBackground(Color.white);
      g2d.clearRect(0, 0, BI_WIDTH, BI_HEIGHT);
      g2d.dispose();

      for (int r = 0; r < COLOR_DIV; r++) {
         for (int g = 0; g < COLOR_DIV; g++) {
            for (int b = 0; b < COLOR_DIV; b++) {
               Color c = new Color((r * 255) / COLOR_DIV,
                        (g * 255) / COLOR_DIV, (b * 255) / COLOR_DIV);
               colorList.add(c);
            }
         }
      }

      imageLabel = new JLabel(new ImageIcon(bImage)) {
         @Override
         protected void paintComponent(Graphics g) {
            super.paintComponent(g);
            paintInLabel(g);
         }
      };
      MyMouseAdapter myMouseAdapter = new MyMouseAdapter();
      imageLabel.addMouseListener(myMouseAdapter);
      imageLabel.addMouseMotionListener(myMouseAdapter);
      imageLabel.setBorder(BorderFactory.createEtchedBorder());

      JButton saveImageBtn = new JButton("Save Image");
      saveImageBtn.addActionListener(new ActionListener() {
         @Override
         public void actionPerformed(ActionEvent e) {
            saveImageActionPerformed();
         }
      });

      JButton clearImageBtn = new JButton("Clear Image");
      clearImageBtn.addActionListener(new ActionListener() {
         @Override
         public void actionPerformed(ActionEvent e) {
            Graphics2D g2 = bImage.createGraphics();
            g2.setBackground(Color.white);
            g2.clearRect(0, 0, BI_WIDTH, BI_HEIGHT);
            g2.dispose();
            imageLabel.repaint();
         }
      });

      JPanel btnPanel = new JPanel();
      btnPanel.add(saveImageBtn);
      btnPanel.add(clearImageBtn);

      setLayout(new BorderLayout());
      add(imageLabel, BorderLayout.CENTER);
      add(btnPanel, BorderLayout.SOUTH);
   }

   private void saveImageActionPerformed() {
      JFileChooser filechooser = new JFileChooser();
      FileNameExtensionFilter filter = new FileNameExtensionFilter(
               "JPG Images", "jpg");
      filechooser.setFileFilter(filter);
      int result = filechooser.showSaveDialog(this);
      if (result == JFileChooser.APPROVE_OPTION) {
         File saveFile = filechooser.getSelectedFile();
         try {
            ImageIO.write(bImage, "jpg", saveFile);
         } catch (IOException e) {
            e.printStackTrace();
         }
      }
   }

   private void paintInLabel(Graphics g) {
      Graphics2D g2d = (Graphics2D) g;
      g2d.setColor(LABEL_DRAW_COLOR);
      g2d.setStroke(LABEL_DRAW_STROKE);
      if (pointList.size() < 2) {
         return;
      }
      for (int i = 1; i < pointList.size(); i++) {
         int x1 = pointList.get(i - 1).x;
         int y1 = pointList.get(i - 1).y;
         int x2 = pointList.get(i).x;
         int y2 = pointList.get(i).y;
         g2d.drawLine(x1, y1, x2, y2);
      }
   }

   private class MyMouseAdapter extends MouseAdapter {

      @Override
      public void mousePressed(MouseEvent e) {
         pointList.add(e.getPoint());
         imageLabel.repaint();
      }

      @Override
      public void mouseReleased(MouseEvent e) {
         Graphics2D g2d = bImage.createGraphics();
         g2d.setColor(colorList.get(random.nextInt(colorList.size())));
         g2d.setStroke(BIMAGE_DRAW_STROKE);
         if (pointList.size() >= 2) {
            for (int i = 1; i < pointList.size(); i++) {
               int x1 = pointList.get(i - 1).x;
               int y1 = pointList.get(i - 1).y;
               int x2 = pointList.get(i).x;
               int y2 = pointList.get(i).y;
               g2d.drawLine(x1, y1, x2, y2);
            }
         }
         g2d.dispose();

         pointList.clear();
         imageLabel.repaint();
      }

      @Override
      public void mouseDragged(MouseEvent e) {
         pointList.add(e.getPoint());
         imageLabel.repaint();
      }
   }

   private static void createAndShowUI() {
      JFrame frame = new JFrame("DrawAndSaveImage");
      frame.getContentPane().add(new DrawAndSaveImage());
      frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
      frame.pack();
      frame.setLocationRelativeTo(null);
      frame.setVisible(true);
   }

   public static void main(String[] args) {
      java.awt.EventQueue.invokeLater(new Runnable() {
         public void run() {
            createAndShowUI();
         }
      });
   }
}