I've just read the Official Twitter Blog
which describes the reason for people seeing authentication errors if they are using the API or plugging into Twitter via a 3rd party website.
Twitter used to operate using two authentication types - namely Basic Authentication
(which stores usernames and passwords) and OAuth
(which just requires your permission, but doesn't store passwords).
To increase security Twitter have done away with Basic Authentication and now support only OAuth
So for anyone seeing the following error: Twitter Error: Basic authentication is not supported
- This is occurring because you are trying to access Twitter using Basic Authentication and this is no longer available, so you will have to adjust your applications to use OAuth instead.