How can I find Mario Maker courses for a specific user?

There is currently no way in game to find a user's courses other than by entering in the Course ID which they may provide. See the online manual. There are, however, alternative means to find a user's courses using their username.

Finding Courses via Miiverse

There is a way to see all of the courses posted by a user within Miiverse. The easiest way to do this is to navigate to their page of posts where you will see all of their recent posts including their Mario Maker stage upload posts. This will provide you with course ID's for their courses which you can use to find at least one course in game. After that you can browse their courses by clicking their user icon in-game.

Using a direct URL with the username

https://miiverse.nintendo.net/users/AnyUserName/posts

Keep in mind that the username is their nintendo network id name, not their display name. For example:

Satoru Iwata has the Display Name "Iwata" but his actual username is "SatoruIwataA". You can see the username beneath their display name on their user page. https://miiverse.nintendo.net/users/SatoruIwataA/posts

Finding a Friend:

1. Open Miiverse.net and log in.
2. Go to your User Page - Click on your user icon in the top left
3. Click on "Friends" (There should be a ##/100 right above it.)
4. Click on your friend's name to visit their page.
5. Click on "All Posts"
6. Scroll through to see all of the stages they've posted recently. You should see a screenshot, the course name, and the course ID.

Finding any other user (Note this is probably easier to use the URL method above)

1. Go to Miiverse.net and log in.
2. Click on the Activity icon at the top
3. Click into the "Search Users" field, type the username, and click the Search icon.
4. Click on the user's name to visit their page. (If you didn't find them, have them double check their name. It should search both their user name and display name.
5. Click on "All Posts"
6. Scroll through to see all of the stages they've posted recently. You should see a screenshot, the course name, and the course ID.

UPDATE:

Finding Friends via the Super Mario Maker Bookmark site

As of Dec. 22 2015, Nintendo created a Bookmark site to help find courses easier. There is a way to get to your friends levels from this site.

1. Navigate to Super Mario Maker Bookmark
2. Log in with your Nintendo Network ID
3. Select your Mii icon in the upper right corner to view your profile
4. On your profile page, you can see "Friends who play this game". You can select any one of them to visit their profile page. From here you can easily Bookmark their stages so that they will be readily available in game.

Please let me know if this answer ever becomes outdated by website changes.