If the client workplace is Manhattan, New York and employee resides in New Jersey, which state's withholding taxes should be deducted? What shall be deduction structure?

Unless the two states have a reciprocity agreement or you are additional registered with the state of New Jersey then withholding is based solely on place of employment. https://www.payroll-taxes.com/articles/state-tax-reciprocal-agreements there is no reciprocity so you do have to deduct taxes based on Manhattan workplace and employee has to file two state income tax returns; they will pay what they owe to New Jersey while waiting on a refund from New York