There are two different ways to view the serial number for a Director appliance. First, you can SSH or console into the Director appliance. The second way is to physically view look for or view the serial number.
Here are some examples of output you may see:
Copyright (c) 1997-2008, Blue Coat Systems, Inc.
Welcome to SG-ME 5.3.1.4
director > sh ver
System version: 5.3.1.4
Build date: 2008/11/21 14:04:34
Build number: 36870
Platform type: 800
Build version: #36870 2008.11.21-140434
Serial number: <not available>
director >
Copyright (c) 1997-2009, BlueCoat Systems, Inc.
Welcome to SG-ME 5.4.1.1 #38189 2009.03.16-092918
director50 > sh ver
System version: 5.4.1.1
Build date: 2009/03/16 09:29:18
Build number: 38189
Platform type: 510
Build version: #38189 2009.03.16-092918
Serial number: xxxxxxxxxx
director >