The only time a browser comes into the equation is at order and collection time. Users do not need to install the certificate for it to change the Unknown Publisher warnings on your downloads, the certificates are already trusted in all versions of Windows (and OSX, most Java devices, Android and iOS too).

Mobile browsers are not supported for ordering.

When ordering the browser does play a key (yet temporary) role because it creates the CSR (certificate signing request), and generates the private key and stores that key temporarily in your browser. The private key not leaving your possession ensures security. Some people might note that with orders from other CAs like Thawte or Verisign that the private key is delivered with the certificate. That is absolutely not secure and, arguably, defeats all of the security of a public/private key encryption scheme since the private key and certificate are transferred over email instead of an encrypted SSL connection (like is done with Comodo).  The recommended browser for ordering is Firefox, because it works out of the box and doesn't prompt users unnecessarily with warnings, prompts and options like some other browsers do. However, most other major browsers will work as well. 

If you've collected your certificate be sure you visit the export instructions at