You can increase security in authentication process by using JWT (JSON Web Tokens) and SSL/HTTPS.
The Basic Auth / Session ID can be stolen via:
By using JWT you're encrypting the user's authentication details and storing in the client, and sending it along with every request to the API, where the server/API validates the token. It can't be decrypted/read without the private key (which the server/API stores secretly) Read update.
The new (more secure) flow would be:
Updated 30.07.15:
JWT payload/claims can actually be read without the private key (secret) and it's not secure to store it in localStorage. I'm sorry about these false statements. However they seem to be working on a JWE standard (JSON Web Encryption).
I implemented this by storing claims (userID, exp) in a JWT, signed it with a private key (secret) the API/backend only knows about and stored it as a secure HttpOnly cookie on the client. That way it cannot be read via XSS and cannot be manipulated, otherwise the JWT fails signature verification. Also by using a secure HttpOnly cookie, you're making sure that the cookie is sent only via HTTP requests (not accessible to script) and only sent via secure connection (HTTPS).
Updated 17.07.16:
JWTs are by nature stateless. That means they invalidate/expire themselves. By adding the SessionID in the token's claims you're making it stateful, because its validity doesn't now only depend on signature verification and expiry date, it also depends on the session state on the server. However the upside is you can invalidate tokens/sessions easily, which you couldn't before with stateless JWTs.