using KeyListener on the KeyEvent.VK_ENTER
Don't use a KeyListener. Swing was designed to be used with Key Bindings. Read the section from the Swing tutorial on How to Use Key Bindings for more information.
set the caret position one line underneath where I add the text to the JTextArea.
Make sure you append a "\n" to the text area when you add the text. Then you can just use:
textArea.setCaretPosition( textArea.getDocument().getLength() );