A Christian comment may be sufficient for your needs. It shows how to use Nancy add-ins for Basic or FormsAuth, and RESTSharp supports Basic auth right out of the box.
I have been working on the Nancy-managed REST API for a long time, we used both RESTSharp and plain HTML + JS as clients, and we decided to implement our own session-based authentication (partly because the add-ons did not exist when we implemented). However, what's nice about this is how easy it is to use, no matter what the client supports. I will quickly explain how this works.
The client sends its username and password (or, if you want, an identifier and a secret key) to create a new session resource using POST /sessions (use HTTPS). This resource contains a session key that can be used for subsequent calls. A session expires after X minutes of inactivity.
Each call made by the service requires a valid session key (other than creating a session). The key is provided either as a cookie or in the query string. When using RESTSharp, we usually set this as a cookie and simply continue to reuse it if it has not expired.
Finally, the session can be destroyed by calling DELETE /session/{key} .
This is a simple but effective way (supposedly HTTPS) to protect the REST API.
Alternatively, you can implement OAuth, which RESTSharp also explicitly supports out of the box.