My button is not gaving out any action when clicked:
(libraryBtn):

package LibraryDatabase;

import static javax.swing.JOptionPane.*;
import java.awt.*;
import javax.swing.*;
import java.awt.event.*;

public class PostLibraryDatabase extends JFrame implements ActionListener {

// Text Fields XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
    JTextField firstName, surName, id, serialNumber, category, dateToday, coverTitle, isbnNumber,
		       authoursName, coAuthoursName, pubDate, loanDate, returnedDate, email, details;
// JButton Fields XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
    JButton writeBtn, displayBtn, exitBtn, libraryBtn, resetBtn ;
    
// ImageIcon Fields XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
    ImageIcon pic = new ImageIcon ("Postlib.png");
		JButton picBtn = new JButton ("" ,pic);

// TextArea Fields  XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
    TextArea information = new TextArea(18, 97);

    DBHandler db = new DBHandler();


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

    public PostLibraryDatabase() {
        setLayout(new BorderLayout());

// text Field


                firstName = new JTextField(16);
                surName = new JTextField(17);
                id = new JTextField(10);
		serialNumber = new JTextField(18);
		category = new JTextField(21);
		dateToday = new JTextField(8);
		coverTitle = new JTextField(25);
		isbnNumber = new JTextField(30);
		authoursName = new JTextField(16);
		coAuthoursName = new JTextField(17);
		pubDate = new JTextField(8);
		loanDate = new JTextField(8);
		returnedDate = new JTextField(8);
                email = new JTextField(27);
                details = new JTextField(48);


                writeBtn = new JButton("Write to database");
                displayBtn = new JButton("Display Records");
                libraryBtn = new JButton("Library Record");
                resetBtn = new JButton("Reset");
                exitBtn = new JButton("Exit");

// position top XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
        JPanel top = new JPanel();
        add("North", top);
// position center XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
        JPanel middle = new JPanel();
        middle.add(new JLabel("First name:"));
        middle.add(firstName);
        middle.add(new JLabel("Surname:"));
        middle.add(surName);
        middle.add(new JLabel("Library ID:"));
        middle.add(id);


        middle.add(new JLabel("Category name:"));
        middle.add(category);
        middle.add(new JLabel("Cover Title:"));
        middle.add(coverTitle);
        middle.add(new JLabel("Today's Date:"));
        middle.add(dateToday);
        middle.add(new JLabel("                            ISBN Number:"));
        middle.add(isbnNumber);
        middle.add(new JLabel("Authours Name(s):"));
        middle.add(authoursName);
        middle.add(new JLabel("                  Co - Authours Name(s):"));
        middle.add(coAuthoursName);
		middle.add(new JLabel("Serial Number:"));
        middle.add(serialNumber);
		middle.add(new JLabel("   Dated Published :"));
        middle.add(pubDate);
		middle.add(new JLabel("      Loan Date:"));
        middle.add(loanDate);
		middle.add(new JLabel("Returned Date:"));
        middle.add(returnedDate);
		middle.add(new JLabel("             Borrowers E-mail Address:"));
        middle.add(email);
		middle.add(new JLabel("Book / Magazine Details:"));
        middle.add(details);


        information.setText(LibraryData.listAll());
        middle.add(information);
        

        add("Center", middle);

// position center
        JPanel bottom = new JPanel();

        bottom.add(writeBtn);
        bottom.add(displayBtn);
	bottom.add(libraryBtn);
        bottom.add(resetBtn);
	bottom.add(exitBtn);

        add("South", bottom);
        add("West", new JPanel());
        add("East", new JPanel());

// ActionListener XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

        writeBtn.addActionListener(this);
        displayBtn.addActionListener(this);
        libraryBtn.addActionListener(this);
        resetBtn.addActionListener(this);
        exitBtn.addActionListener(this);

// Windows Size  XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
        setSize(730, 690);
        top.add (picBtn);
        
        setTitle("Post Library Database");

        setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
        setVisible(true);
        setResizable(false);
    }
//  Action Performed XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
    public void actionPerformed(ActionEvent e) {
        if (e.getSource() == writeBtn) {
            String f = firstName.getText();
            String s = surName.getText();
            String z = id.getText();
            String a = serialNumber.getText();
            String b = category.getText();
            String c = dateToday.getText();
            String d = coverTitle.getText();
            String g = isbnNumber.getText();
            String h = authoursName.getText();   
            String i = coAuthoursName.getText();
            String k = pubDate.getText();
            String l = loanDate.getText();
            String m = returnedDate.getText();
            String n = email.getText();
            String o = details.getText();


// XXXXXX if any field is blank, signal an error XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
            if (f.equals("") || s.equals("") || i.equals("")) {
                showMessageDialog(this, "One or more fields blank");
                return;
            }
            boolean ok = db.write(z, s, f, a, b, c, d, g, h, i, k, l, m, n, o);
            id.setText("");
            if (!ok) showMessageDialog(this, "Duplicate key " + id);
            else {
                firstName.setText("");
                surName.setText("");
                serialNumber.setText("");
                category.setText("");
                dateToday.setText("");
                coverTitle.setText("");
                authoursName.setText("");
                coAuthoursName.setText("");
                pubDate.setText("");
                loanDate.setText("");
                returnedDate.setText("");
                email.setText("");
                details.setText("");
            }
        
            if (e.getSource() == libraryBtn) {
                 new CheckLin();
             

	} else if (e.getSource() == exitBtn) {
            LibraryData.close();
	    System.exit(0);
			}

		   }
		}
	}

It seems that you have messed up with the brackets.
You have this:

public void actionPerformed(ActionEvent e) {
        if (e.getSource() == writeBtn) {
        
            if (e.getSource() == libraryBtn) {

            } else if (e.getSource() == exitBtn) {

	    }

       }
}

Maybe you need this:

public void actionPerformed(ActionEvent e) {
        if (e.getSource() == writeBtn) {
             
	} else if (e.getSource() == libraryBtn) {

        } else if (e.getSource() == exitBtn) {

	}
}

Can you please verify it?

(1) In void actionPerformed(ActionEvent e)method body there are missing code blocks for two cases:
if (e.getSource() == displayBtn){...} and
if (e.getSource() == resetBtn){...}
(2) in void actionPerformed(ActionEvent e)method body
one curly bracket '}' is missing in line 178
That's why, I guess, the buttons do not work properly.

While debugging I'd add a trailing else:

} else if (e.getSource() == exitBtn) {
 
	}else {
            System.out.println("unhandled action " + e); // tell em about it
        }
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