Since NH and MA do not have reciprocity with other states, taxes should be withheld and paid to the state in which they live and the state where they work. You would still need to determine which state had the higher rate. But, the employee is required to file with their resident state and non-resident state if they earn income in the non-resident state.
Both states will penalize the employees for not having enough paid in during the year. So, based on the withholding rates for each state, you would look at what the home state tax would be on the income, then look at the non resident state tax would be. If the home state tax is greater, then you would pay the nonresident state their rate, then pay the resident state the difference. This is what would keep the employee from having an underpayment penalty.
If the tax rate for the nonresident state is higher than the resident state, then their would be no tax liability to the resident state, so you would only withhold for the nonresident state.
MA Multiple State Withholdings
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