C# Method to determine database status
Closing the database connection in the finally will cause an exception if the Open () call throws an exception, so you better close the connection  immediately after you opened it in the try block.  

To overcome this problem in a clean way you should let the using block handle it to close the database connection.



Method's should be named using a verb or verb phrase. So better name it e.g IsDatabaseServerAlive ().

Otherwise it looks ok for me.