I have a Drupal install where PHp runs through fastcgi.

Here I describe how I got this module to work.

1) webdav.module uses PHP_AUTH_USER and PHP_AUTH_PW to do basic auth.

These headers aren't available on fastcgi. To make them available, I added

# To get webdav working
RewriteRule .* - [E=HTTP_AUTHORIZATION:%{HTTP:Authorization}]

directly after "RewriteEngine on" in .htaccess.

Then in in the module I added

if (isset($_SERVER['REDIRECT_HTTP_AUTHORIZATION'])) {
list($_SERVER['PHP_AUTH_USER'], $_SERVER['PHP_AUTH_PW']) = explode(':' , base64_decode(substr($_SERVER['REDIRECT_HTTP_AUTHORIZATION'], 6)));
}

just before the comment " // Lets get authentication stuff"

Then it seems to work.

I am not sure if the module should be patched to allow fastcgi setups, but this should at least be in the documentation.