Knowing which dog might be prone to have this problem is certainly not an exact science! Dogs which lack self-confidence, dogs that have been adopted from a pound, dogs that from birth have always been with people and suddenly are alone all of these situations could cause separation anxiety.
Diagnosing S.A. (separation anxiety) can be slightly easier. The dog will more than likely bark excessively during the time it is alone. There will probably be destructive behavior, i.e. chewing, tearing up things, etc. Also, the owner will notice that the dog is eliminating in the house. A normally happy dog might become sad or depressed, and his appetite will decrease. If the dog is crated, he might try to escape from the crate. Dogs suffering from S.A. have been known to totally destroy a crate, and in doing so hurt themselves quite seriously.
All in all, this may be one of the most difficult problems a dog owner can face. Curing this behavior takes much patience, fortitude and, most of all, love. To read one of the most touching stories I have ever read on this subject, please read the story of Midnite. Another article titled Alone, But Not Lonely talks about dealing with separation anxiety.