Encountered a problem accessing my tracks from browser. Usually i type this link in my browser to access downloadable tracks:
http://api.soundcloud.com/tracks/294324
As no one else has mentioned it, in Google Chrome under DevTools F12, you can grab your client_id from the Network tab on reload. Many items listed here reference the SoundCloud API and require your client_id to access it. Simply click any item in the list that isn't an image/base64 item (like something that begins with id?...), then look in the Request URL for the client_id.
It's the simplest way to grab the ID; now that you know how, don't do anything illegal k thx.
For those thinking this is the same as the other answer mentioning a download link, this method skips that entirely as many songs are no longer downloadable from SoundCloud itself, but from external sources that are linked instead.