Update Sep. 2016: there are a few tools created by the community that can do this for you:

Git doesn't support this, but Github does via SVN. If you checkout your code with subversion, Github will essentially convert the repo from git to subversion on the backend, then serve up the requested directory.

Here's how you can use this feature to download a specific folder. I'll use the popular javascript library lodash as an example.

1. Get the repo URL. First, copy the URL of the Github repo to your clipboard. 

2. Modify the URL for subversion. I want to download the folder at /docs from the master

   branch, so I will append trunk/docs. Full URL is now https://github.com/lodash/lodash/trunk/docs. See my note below for a more in-depth explanation of why we must use this URL format.

3. Download the folder. Go to the command line and grab the folder with SVN.  svn checkout https://github.com/lodash/lodash/trunk/docs

You might not see any activity immediately because Github takes up to 30 seconds to convert larger repositories, so be patient.

Full URL format explanation:

• If you're interested in master branch, use trunk instead. So the full path is trunk/foldername
• If you're interested in foo branch, use branches/foo instead. The full path looks like branches/foo/foldername
• Protip: You can use svn ls to see available tags and branches before downloading if you wish

That's all! Github supports more subversion features as well, including support for committing and pushing changes.