I have only done indoor remodeling in fall and winter, so it never occurred to me that it would be a problem.

but don't want the dust and smells to drive us crazy with us all inside

You can avoid a major source of dust and overall messiness by using non-load bearing metal framing instead of wood. It also goes up much faster than wood, which should be a cost savings. Don't let some sharp contractor moan on about how light steel is more complicated and time consuming to install!

Talk to the contractor about ventilation, sealing off the construction area, and how your HVAC system works (if it's forced air, you'll need to change the filters often). How easy will it be to open all the basement windows on the warmer days? An hour or two of cross ventilation with a commercial fan blasting out one window may be possible if the door to upstairs can be adequately sealed off to avoid removing all your heated air upstairs. If the NJ winter is cold and snowy for months without a break, that won't be practical.

As apple pie mentioned as much as possible close off the work area , the metal studs are fine but I don’t like them where I want to hang art or speakers so keep that in mind.If there is a walk out from the basement all the better .I have often done renos for clients in the winter because in the summer they want to be outside so this seems pretty normal to me.

Doing a finished basement you will not be smelling anything bad if you smell anything at all.

Dust is very limited, drywall sanding is usualy wet or with a vacuum dust attachment, etc.