Unless Anthem is going to back date the effective date of the provider's contract to cover the dates of service that the patient has received checks for, then yes, the patient needs to cash the checks and if she hasn't already paid your office pay you what she owes.
If Anthem is going to back the effective date to cover all of her visits or to some degree that would cover some of her visits, then it is likely easier for her to not cash the checks. I work for an insurance company and it easier for us if the patient hasn't cashed the check because we can put a stop pay on the check and then reprocess the claim to issue the payment to the provider.