Is verbal permission enough to open a safe?
So long as your mother is mentally competent, she can designate verbally who she wishes to act on her behalf and it appears that she has determined that your younger sister is the person she wants to act on her behalf at this time. However, if your mother were to become mentally incompetent, her verbal authority to your sister to be able to act on her behalf would be totally ineffective.
The best course of action would be for all of the family to meet and for you to voice your concerns to your mother as to how her financial affairs are being managed and should be more effectively taken care of in the future. Your mother should have a lawyer prepare a Durable Power of Attorney for her. This is a document that would be effective both now and in the future if she becomes incapacitated that authorizes someone or multiple people to act on her behalf in regard to financial matters. The authority granted in the Durable Power of Attorney can be as broad or as limited as she wants.