I installed recaptcha on my local Dev using these steps, so this may help you.
NOTE: there are some intermediate steps that may or may not effect other parts of your site, so it is best to test this first. Also, make sure you have the appropriate CMS version. I believe the minimum is 2.3.2.
1. Download and install from Modules: SpamProtector
1b. extract or copy the folder "spamprotector" into your CMS root. You may need to rename the folder slightly by taking off the version number.
1c. rebuild the database
2. Download and install from Modules: ReCaptcha
2b. extract or copy the folder "recaptcha" into your CMS root. You may need to rename the folder slightly by taking off the version number.
2c. rebuild the database
3. For the firefox fix, add a closing tag for the input HTML (around line 171) in file: recaptcha/code/RecaptchaField.php
3b. For the FF fix, remove the xml declaration at the top of the templates, page.ss and make sure it is declaring xhtml: <?xml version="1.0" encoding="UTF-8"?>
4. Go to http://recaptcha.net/ and log in OR sign up for a new account.
4b. Create keys for your site's domain.
4c. Insert the code for the keys into the ReCaptcha _config file. i.e.
RecaptchaField::$public_api_key = '<your public key>';
RecaptchaField::$private_api_key = '<your private key>';
5a. In your mysite _config file, add the following line:
5b. Visit the page where your form is and ?flush=1. You should see the working recaptcha field amongst the default form.
Hope this has helped.