Is your question about whether you can open a 10.1 geodatabase with ArcGIS Desktop 9.3.1?
If so the answer is no. ESRI's stance until recently has been one of backwards-compatibility only. Meaning the client version must be at or newer than the geodatabase release version. See the compatibility table here: Client and geodatabase compatibility
Interestingly, at 10.1, they added forwards-compatibility for ArcGIS 10.0 clients to be able to access ArcGIS 10.1 geodatabases with some limitations.
See the "Interrelease compatibility support" section of this help topic: What's new for geodatabases in ArcGIS 10.1
Note that you can create previous release version personal and file geodatabases from newer versions using geoprocessing tools:
Also older clients can in general consume services from newer versions of ArcGIS Server (link), although documentation for this is sparse. I could find no "ArcGIS Server 10.1 compatibility matrix", for example.