February Due Date and Flu Shot

So I am due February 5th, but of course it could be mid-late January.

At my appointment tomorrow I will ask if they recommend me getting the flu shot...but do they offer them to babies? I wouldn't want my baby to get one.

Also everyone else in my house (two parents and two brothers) I'll assume won't be getting one as they never do which I just thought about but now I don't know if I'm comfortable living with a bunch of people unvaccinated/handling the baby or touching everything I'm touching (household items) with a newborn born in flu season. Or am I being hyper sensitive.