You will need to download the most recent version of Connector/J from the MySQL website. This is the official JDBC driver for MySQL, made available under the GPL.
Once you have the driver unarchived, move the .jar file somewhere permanent. I like to put all my third party JDBC drivers together in the SQL Developer root directory.
Next go to Tools > Preferences > Database > Third Party JDBC Drivers. click Add Entry and browse to the jar file.
You should now find a new tab has appeared when you next create a new database connection.