As long as you download the files using classic FTP, and open/edit the files in a text or HTML editor only (no WYSIWYG/preview function!) this is totally harmless. Looking at the file in a text editor isn't dangerous, executing it in a browser is.
Get the latest security updates for the browser you will be using to look at the pages - through Windows Update for IE, or by just downloading the latest Version of Firefox, Chrome or whatnot. I'd recommend Firefox because of its Web Developer toolbar.
To be 100% safe during editing in a WYSIWYG editor, and to test the cleaned pages in a browser (if you have a local Joomla to test), you could disconnect the computer from the Internet while editing.
Also, having a virus scanner running in the background is not a bad idea.
However, don't forget to fix the actual problem first - the vulnerability that made the injection possible! I assume from what you're writing that that has been already sorted but make sure you find out where the break-in occurred.
As a minimum measure, change all passwords to all web hosting related accounts (FTP, Control Panel, etc.).
The Google Webmaster Blog's article My Site's been hacked - now what? is always good reading as well. It also describes how to get the site re-indexed with Google fast.