Christmas in Paris

My daughter and I are fulfilling a dream of going to Paris together -- and she has suggested this Christmas. Several years ago she and I spent the holidays in London and we both agree that it was a magical and very special trip for both of us. While we have a pretty good idea of the things we'd like to do (both of us are teachers, and I have been taking students abroad -- with my daughter accompanying us -- since she was six -- so almost 30 years of travel!) -- but have no idea what are the better places to stay. We don't need (nor can we really afford) a luxury hotel, and in fact, many times, when she and I have traveled in the US and in the UK< we have stayed in hostels. We are also trying to stay within a certain budget -- as I retire this year, and she is a newlywed! So can anyone offer any viable suggestions? I am thinking that we'd like to spend no more than 200 per night... and we were thinking of Montmartre -- but I don't know about safety there... We are both quite fit -- she is a yoga instructor, and I practice regularly -- so walking is not a particular issue (although I AM 67. and beginning to tire more quickly :)) I've been looking at VRBO and AirBNB as well as Booking.com. I am just not sure as to the safety issue... Citiidine Montmartre was recommended, as were several hostels in that area -- but again -- I am not familiar enough with the neighborhoods in Paris to know if they would work.
I would say that our travel "philosophy" is to live like a local as much as possible. Walk, use public transport. eat at local eateries, shop in the market, etc. Of course we need to see the sights -- but it's the experience of BEING there that we are looking for. The last time I was in Paris was about 25 years ago on a training tour with a student travel company, which is certainly no help at all. So I am relying on you good people to give me some guidance! Hoping to hear something soon, as I'm told we should book EARLY.
Merci beaucoups in advance!!