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Your spouse would be considered a resident of Minnesota like yourself, assuming she did not elect out of that. So, you would typically file married-joint in Minnesota and New York, as you can't file married-separate in New York if you file married-joint on your federal return.
Your income tax rate in New York will be reduced based on the income you earn out of state, which is your income. So, you shouldn't end up being double taxed.
You need to look for a credit for taxes paid to another state on your Minnesota return so you won't get taxed twice on your spouse's New York income. You should prepare your New York return first in TurboTax, before you prepare the Minnesota return.
You file as non-residents in New York.
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