Jessica Fletcher is a best selling mystery novelist with her picture on the dust jacket of her books, making her quite famous, as has been established in previous episodes. Yet, she is able to go undercover to this resort full of wealthy people and no one recognizes her.
In the end, Jessica finds out that Mr. Greenberg has the only master key to the safe deposit boxes, then explains how she knew he was the killer/thief because Alva's safe deposit key was kept in her handbag, but was found on a bureau in her room when she was found dead, so he must have used it and returned it. He wouldn't need the key at all if he had the master key.