Employers verify work authorization from documents the employee presents, not from a family relationship or a description of the principal's visa. Plan on the spouse holding a passport, the admission record, and whatever notation or evidence establishes the employment-authorized E dependent status, and confirm what that is once admission has actually occurred.
Check the admission record before the start date
After entry, retrieve and read the I-94 and any classification notation rather than assuming the visa foil settles it. If the classification or expiry date is not what was expected, that is worth raising immediately, since a hiring manager will not be able to interpret it. Keep the marriage certificate and the children's birth records with the file too, translated where necessary, because relationship evidence supports the dependent applications themselves.
Prepare the spouse for the practical mechanics as well as the documents. Hiring paperwork is completed within a short window after employment begins, and the employee presents documents from a defined list rather than explaining a family relationship. Knowing in advance which item will be presented, and having it physically to hand, removes the most common cause of a delayed start.
Where the admission record does not show what was expected, that is a question for counsel immediately rather than something to raise with the employer, who has no way to resolve it and every reason to be cautious.