I'll answer your question as though it were not your wife but a stranger. If it were a stranger, you would go to the police and have criminal charges filed.
I suggest you buy a subscription to one or more of the credit reporting agencies. They will advise you by email when something has happened on your credit. That will alert you in the beginning and you can put a stop to it right away.
That won't solve your primary problem - If you and your wife are together and you are unwilling to charge her with a crime, you have a much larger problem that is beyond what I can provide here.