There is no way to answer that without knowing a lot of details about each of your tax situations. Assuming you both live together with your baby, you could each see what your refund is without claiming the baby and then each enter the baby to see which of you it benefits most. Make sure you've both entered all your other tax information before making the comparison. Whoever benefits the most would probably want to claim the baby as their dependent. Whoever does not claim the baby needs to make sure they delete the baby from their return before they file it.
When answering the question about how long the baby lived with you, the correct answer is all year. If you don't say all year, TurboTax will disallow the baby as a dependent. All year is the right answer for a child born during the year.
