I take it you have seen your NJ return, because tax due could be due to many things.
But, yes, I would think you would want to adjust your NJ return for the HSA contributions.
May I assume that State Wages (box 16 on your W-2) shows the Box 1 Wages plus the code W amount? This would be what the starting point would be for NJ Gross Income.
If so, when you get into your NJ interview for a non-resident return, you should pretty quickly come to a screen entitled "Other Wage Adjustments - W-2".
In the detail boxes below, enter something like "HSA non-resident adjustment", and a negative number.
The negative number should be the amount of HSA contributions not attributable to New Jersey. You can calculate this by taking your code W amount (i.e., your total "employer contributions") times the ratio of your NJ wages divided by your total W-2 wages, and then subtracting the result form the code W amount. This should result in Gross Income for NJ not including the code W amount, except for the small amount attributed to NJ.
See if this gets you to a more appropriate result.
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