Based on visualstudio.com, it looks like you're connecting to hosted TFS.
You can enable alternate credentials and then use those credentials to auth via basic auth.
If you go to your profile in the web page for TFS (upper right hand corner in hosted), there's a credentials tab. enter a username and password for alternate credentials and you can now send via basic auth header.
Programmatically in C# it's:
NetworkCredential netCred = new NetworkCredential(
"altUserName",
"altPassword");
BasicAuthCredential basicCred = new BasicAuthCredential(netCred);
Buck blogged about it here: