Assets that are held in joint names generally pass to the surviving joint owner upon the death of the other joint owner. This means that when your mother passes away,
whatever assets remain in the various joint accounts would become payable to you alone -- you would have no difficulty getting access to the funds.
Joint property isn't generally subject to the jurisdiction of the Probate Court. All you would have to do is just withdraw the funds from the various accounts. However, there is one exception to this rule. If your mother designated you as a joint owner merely for convenience purposes and did not intend to have the assets pass to you upon her death, the joint accounts would be part of her estate and your stepbrother would have a claim for half of his mother's joint accounts.
It would be helpful for her to write a document that indicates what her intention is as to the joint accounts. Another point to keep in mind: if she requires long-term care in a nursing home in the future, the joint checking, savings and CD accounts would all be considered her assets if she were to apply for Medicaid.