1. Edit my.ini in Windows (or /etc/my.cnf in Linux) in of the mySQL server.
   1. Add old_passwords=1 to [mysqld] section (UPD. Not necessary!)
   2. Add skip-secure-auth to [mysqld] section
   3. Restart MySQL service (in Windows) or execute /etc/init.d/mysqld restart (on Liniux).
       
2. Connect to mySQL server under the root and execute some following commands:
   1. (Not necessary but just to make sure that previous steps was successful, and to make sure that old-style passwords of MySQL version prior to 4.1 is allowed)
      SHOW VARIABLES LIKE "%password%";
   2. USE mysql;
      
   3. SET password FOR 'username'@'host'=OLD_PASSWORD('password');
      
   4. UPDATE user SET plugin='mysql_old_password' WHERE user="username" AND host="host";
      
   5. FLUSH privileges;

That’s all! Try to connect to MySQL database with your old MySQL-client. If you will get errors like

client does not support authentication protocol requested by server; consider upgrading mysql client

then you probably did something wrong in previous steps.

You may also download Zeos Database Explorer 1.5.21 from attachment, or even its source code in Delphi. Zeos Library available at SourceForge.