Avoid December and January at all costs if you don't want to pay triple for everything. It's peak season, it's pretty hot and humid, and prices are through the roof. August and September aren't great either because it's rainy season and there's a higher risk of hurricanes in the region. April and May get pretty sticky with the weather too.
Your best bet is to go between February and March, or in October and November. In those months the weather is way more manageable, it rains less, and prices drop quite a bit compared to the holidays. February and March especially are good because the peak of summer is over and the city isn't so packed. November is decent too, though prices start to creep up a bit heading into the holidays.
What makes sense is to avoid end-of-year school vacations if you can, and try to travel during those periods when most people are busy with other stuff. With your family you'll be way more comfortable, hotels and restaurants won't be so crowded, and your wallet will thank you. Check out mid-week flights - there are usually good deals on those.