After updating to OSX Mavericks, WebDAV no longer works properly. There's currently no solution available. Apple has released a workaround while they are attempting to fix the problem with the next update.

Workaround

To workaround the issue, download the 3 files at this location:

http://www.opensource.apple.com/source/webdavfs/webdavfs-334.2/webdav_cert_ui.tproj/English.lproj/MainMenu.nib/

These 3 files will have to be placed in the folder:

/System/Library/Filesystems/webdav.fs/Support/webdav_cert_ui.app/Contents/Resources/English.lproj/MainMenu.nib

You would normally overwrite the current ones present, but there's a catch. You can't simply overwrite the files because they are still in use by the system. Instead, create a folder called with an alternative name (e.g.: MainMenu) inside /System/Library/Filesystems/webdav.fs/Support/webdav_cert_ui.app/Contents/Resources/English.lproj/.

Place the 3 files inside the newly created folder.

In the same folder where you created MainMenu, remove the original folder MainMenu.nib, immediately followed by a rename of MainMenu to MainMenu.nib. Then restart the system.