On the NetTuts.com site there's a tutorial posted showing you how to set up and use basic authentication in the WordPress REST API. This is part two in their series introducing the WordPress REST API.

In the introductory part of this series, we had a quick refresher on REST architecture and how it can help us create better applications. [...] In the current part of the series, we will set up a basic authentication protocol on the server to send authenticated requests to perform various tasks through the REST API.

They talk about the methods that are available for authentication and how to configure your server and WordPress instance to use it. From there they show how to make authenticated requests to the API using various tools:

• Postman
• a Javascript framework (jQuery)
• the command line via curl
• using the WP HTTP API

Example code and screenshots are provided for each (where appropriate) helping to ensure you're up and working quickly.