By default both versions are sent.
It would require an add-on to prevent it.
To do it manually, you could edit the email templates and replace the HTML content with the plain text version.
However, in order to do that, you would need to enable debug mode and do it for each email template.
Bear in mind that any templates edited manually would revert on an upgrade, unless you did it as email template mods in the form of an add-on.
I just wanted to say Thanxx @Mayhem30 for mentioning this! I tried editing the text version in the phrase manager, adding a 'You can also Contact Us if needed' note and it just would not show up in the (html) email. And now, after reading your & @Brogan's discussion here I understand why!