No new mail notification at apple mail dock icon when rules apply

Apple Mail rules run automatically only the first time they're set. After that rules will only run manually.

Going into Preferences and unchecking 'Active' will deactivate the rule and will never run it.

This is an edited version — quite possibly not the best one — of the script shown in this answer to the Automatically Save Attachments in Mail.app in 10.8 Mountain Lion thread, I've tested it and it works:

using terms from application "Mail"
    on perform mail action with messages theMessages for rule theRule
        tell application "Mail"
            repeat with eachMessage in theMessages
                repeat with theAttachment in eachMessage's mail attachments
                    set attachmentName to name of theAttachment
                    if attachmentName is "blank expense report" then
                        set savePath to "Macintosh HD:Users:yourUsername:Inbox:" & attachmentName
                        try
                            save theAttachment in file (savePath)
                            on error err
                            display display "The Following Error Occurred:" & err
                        end try
                    end if
                end repeat
            end repeat
        end tell
    end perform mail action with messages
end using terms from

This script will save the attachment named "blank expense report" in a named "Inbox" at Machintosh HD:Users:yourUsername:Inbox. Be sure to change the value of savePath variable accordingly to your needs.