Here are a few things you might try to get a better understanding of the problem:
If you open up Developer Tools or Firebug in your browser and go to the MODX file manager, you should see a request for connectors/browser/directory.php. The request and response might give you some clues about what's going wrong. Normally the request to directory.php returns a listing of subdirectories in JSON format.
Can you upload a file? If so, FTP or SSH to your server and see where the file ended up.
Check MODX_ASSETS_PATH in core/config/config.inc.php. Make sure it's right and that the user PHP is running as has read/write/execute permissions on that directory. On the subject of PHP, if you go to Reports > System Info and view phpinfo(), what does it say the Server API is? If it says 'Apache 2.0 Handler' then you'll likely have to set 777 permissions on that directory.
Create a new directory in your web root via FTP. Say, 'testdir'. Put an image in it and make sure you can see it in your browser (
http://yoursite.com/testdir/someimage.jpg ). Make sure the directory has the proper permissions for whoever PHP is running as. Create a Media Source for the new directory:
basePath: testdir/
basePathRelative: Yes
baseUrl: testdir/
baseUrlRelative: Yes
Go to the file browser and switch to the new media source. Does it have the same problem?