Here is just a small SSCCE:

import java.awt.*;
import javax.swing.*;

public class TextAreaInScrollpane
{
    static JPanel mPanel = new JPanel();
    static JScrollPane mScrollPane;

    public static void main(String[] args)
    {
       mPanel = new JPanel(new GridBagLayout());
       GridBagConstraints constraints = new GridBagConstraints();
       constraints.gridwidth = GridBagConstraints.REMAINDER;
       constraints.anchor = GridBagConstraints.NORTHWEST;
       constraints.fill = GridBagConstraints.BOTH;
       constraints.weightx = 1.0;
       constraints.weighty = 1.0;

       mScrollPane = new JScrollPane(mPanel);

       JTextArea ta = new SpecialTextArea();
       ta.setLineWrap(true);
       ta.setText("Hallo, mein Name is Theodor. Hallo, mein Name is
Theodor. Hallo, mein Name is Theodor. Hallo, mein Name is Theodor.
Hallo, mein Name is Theodor. Hallo, mein Name is Theodor. Hallo, mein
Name is Theodor. Hallo, mein Name is Theodor. Hallo, mein Name is
Theodor. Hallo, mein Name is Theodor. Hallo, mein Name is Theodor. ");
       mPanel.add(ta, constraints);

       JFrame frame = new JFrame();
       frame.setDefaultCloseOperation( JFrame.DISPOSE_ON_CLOSE );
       frame.setContentPane( mScrollPane );
       frame.pack();
       frame.setVisible(true);
    }

    @SuppressWarnings("serial")
    private static class SpecialTextArea extends JTextArea
    {
        public SpecialTextArea()
        {
            super("");
            super.setEditable(false);
            super.setLineWrap(true);
            super.setWrapStyleWord(true);
        }

        public Dimension getMinimumSize()
        {
            return new Dimension(1,1);
        }
    }
}

The text area isn't going to get smaller if it is in a scroll pane.  The
frame will but that's the scroll pane's job to make the underlying
components still fit.  You do have some other problems though when you
make the frame bigger because you have GridBagConstraints.fill set to
BOTH it makes the text area larger.  You have a conundrum going on.  Use
a different layout manager or set the preferred size of the text area
and turn off the fill or take it out of the scroll pane or ...  There
are too many options to give you without knowing exactly what you want
to do in the end.