Internal padding for JTextArea with background Image

Use a custom border that extends AbstractBorder. Something like this:

Getting the exact shape & color is left as an exercise for the reader. :)

import java.awt.*;
import java.awt.geom.*;
import javax.swing.*;
import javax.swing.border.AbstractBorder;

class TextBubbleBorder extends AbstractBorder {

    private Color color;
    private int thickness = 4;
    private int radii = 8;
    private int pointerSize = 7;
    private Insets insets = null;
    private BasicStroke stroke = null;
    private int strokePad;
    private int pointerPad = 4;
    RenderingHints hints;

    TextBubbleBorder(
        Color color) {
            new TextBubbleBorder(color, 4, 8, 7);
    }

    TextBubbleBorder(
        Color color, int thickness, int radii, int pointerSize) {
            this.thickness = thickness;
            this.radii = radii;
            this.pointerSize = pointerSize;
        this.color = color;

        stroke = new BasicStroke(thickness);
        strokePad = thickness/2;

        hints = new RenderingHints(
            RenderingHints.KEY_ANTIALIASING,
            RenderingHints.VALUE_ANTIALIAS_ON);

        int pad = radii + strokePad;
        int bottomPad = pad + pointerSize + strokePad;
        insets = new Insets(pad,pad,bottomPad,pad);
    }

    @Override
    public Insets getBorderInsets(Component c) {
        return insets;
    }

    @Override
    public Insets getBorderInsets(Component c, Insets insets) {
        return getBorderInsets(c);
    }

    @Override
    public void paintBorder(
        Component c,
        Graphics g,
        int x, int y,
        int width, int height) {

        Graphics2D g2 = (Graphics2D)g;

        int bottomLineY = height-thickness-pointerSize;

        RoundRectangle2D.Double bubble = new RoundRectangle2D.Double(
            0+strokePad,
            0+strokePad,
            width-thickness,
            bottomLineY,
            radii,
            radii
            );

        Polygon pointer = new Polygon();

        // left point
        pointer.addPoint(
            strokePad+radii+pointerPad,
            bottomLineY);
        // right point
        pointer.addPoint(
            strokePad+radii+pointerPad+pointerSize,
            bottomLineY);
        // bottom point
        pointer.addPoint(
            strokePad+radii+pointerPad+(pointerSize/2),
            height-strokePad);

        Area area = new Area(bubble);
        area.add(new Area(pointer));

        g2.setRenderingHints(hints);

        Area spareSpace = new Area(new Rectangle(0,0,width,height));
        spareSpace.subtract(area);
        g2.setClip(spareSpace);
        g2.clearRect(0,0,width,height);
        g2.setClip(null);

        g2.setColor(color);
        g2.setStroke(stroke);
        g2.draw(area);
    }

    public static void main(String[] args) {
        SwingUtilities.invokeLater(new Runnable() {
            public void run() {
                JLabel l = new JLabel(
                    "The quick brown fox jumped over the lazy dog!");

                l.setBorder(new TextBubbleBorder(Color.MAGENTA.darker(),2,4,0));
                l.setOpaque(true);
                l.setBackground(Color.BLACK);
                JOptionPane.showMessageDialog(null, l);
            }
        });
    }
}

You should use an Border for that, more specifictly you should use BorderFactory.createEmptyBorder(int top, int left, int bottom, int right):

textArea.setBorder(BorderFactory.createEmptyBorder(10,10,15,10));

You should also override paintComponent instead of paint. Also, use setRows() and setColumns() to set the size of the textArea, then you can use pack() instead of setSize(400,400) which is not recommended. See this example:

import java.awt.BasicStroke;
import java.awt.BorderLayout;
import java.awt.Color;
import java.awt.Graphics;
import java.awt.Graphics2D;

import javax.swing.BorderFactory;
import javax.swing.JFrame;
import javax.swing.JPanel;
import javax.swing.JTextArea;


public class Test extends JFrame {

    class MyTextArea extends JTextArea {

        public void paintComponent(Graphics g) {
            super.paintComponent(g);
            Graphics2D g2 = (Graphics2D)g;
            g2.setColor(Color.PINK);
            g2.setStroke(new BasicStroke(4));
            g2.drawRoundRect(3, 3, getWidth()-7, getHeight()-7, 5, 5);
        }

    }

    public Test() {
        JPanel panel = new JPanel(new BorderLayout());
        JTextArea textArea = new MyTextArea();
        textArea.setRows(3);
        textArea.setColumns(25);
        textArea.setBorder(BorderFactory.createEmptyBorder(10, 10, 10, 10));
        panel.add(textArea, BorderLayout.NORTH);

        add(panel);
        pack();
        setDefaultCloseOperation(EXIT_ON_CLOSE);
        setVisible(true);
    }

    public static void main(String[] args) {
        new Test();
    }

}

To post code, indent every line by four spaces.

I assume you are overriding paintComponent()* for your JTextArea. If you are, make sure that it is transparent by adding

setOpaque(false);

*This also works if you override paint(), but as trashgod correctly states, that would interfere with paintBorder().