This doesn’t sound normal to me. I have a Drobo connected to a Mac Mini running 10.6.4 when I sleep or shutdown the Mac the drobo goes into standby mode (all lights off except for the small orange light which you have to remove the front cover to see.)
Try shutting down the Mac and then disconnecting the data cable for the drobo wait a few min and see if the drobo goes into standby. If it still does not, as long as the computer is off you should be safe to unplug the drobo’s power wait about 30sec and then plug the power back in. Reconnect the data cable, Sart the computer back up and see if the drobo will now sleep and wake along with the Mac.
On a side note while the drobo does sleep and wake with my Mac it rarely will appear back in the dashboard after waking the Mac from sleep. A restart will generally cause it to appear correctly again. This is just a minor annoyance however since the dashboard software is not necessary to use the drobo.