I want to add a captcha to the PDF generator from the Print Module so that my server doesn't waste time generating a PDF for robots and to prevent DOS attacks.
There is a similar question asked, but the accepted answer doesn't actually answer my question (the PDF Version part), but rather only the E-mail part.
Since the PDF Version doesn't link to a form, there isn't a form ID I can add to the Captcha module. Is there a way to require a CaPTCHA to download/generate a PDF?
First you should avoid captcha, their are not user friendly. But good new, in this case you don't need it :).
Print module generate the pdf once and cache it https://drupal.org/node/516466#comment-6390678. So DOS are not possible on pdf generation.
You can still have DOS by requesting a pdf thousand of time, saturating your bandwidth. If you are really worried about that you can for example configure Apache with mod_evasive http://systembash.com/content/how-to-stop-an-apache-ddos-attack-with-mod_evasive/
By robot I suppose you mean search engine robot. You can add that in your robot.txt (But I think is not necessary):
User Agent: *
Disallow: /*.pdf$
