First of all you should consider NOT doing basic authentication directly - but rather use the OAuth2 authorization server approach - read this first:
http://leastprivilege.com/2013/11/13/authorization-servers-are-good-for-you-and-your-web-apis/
If you still want to do basic auth - then in this repo you will find an implementation for Katana:
nuget: Thinktecture.IdentityModel.Owin