Hello. I live there and personally I would recommend Quartier Du Marais (3rd arrondissement and 4th arrondissement)
It's full of life every day, a lot of small businesses and great restaurants nearby. Moreover, you have a lot of metro stations around which you will often take to visit the city. I think you can find a nice apt on Airbnb.
Here is a link of the tourist info Paris about the Marais :

http://en.parisinfo.com//transports/73274/Quartier-du-Marais

I've rented apartments in Paris about 7 times and 6 of them have been on the left bank...cheaper and more interesting. I find grocery stores and other conveniences are more readily available int he 5th and 6th arrondissement. I stayed near the Grand Palais and just found it more expensive and harder to find grocery stores. Check out my website: www.francetraveltips.com

A few years ago I stayed 3 weeks in Montmartre and I absolutely loved it. I stayed at Citadines apart-hotel near the Boulevard Clichy

I've lived in multiple neighborhoods across Paris in nearly 4 years and the best one, without a doubt, was in the 4eme on Ile Saint-Louis and the nearby Quartier Du Marais . 

Generally I do not recommend staying in the 15/16/17eme, they're more residential and you'll spend valuable time commuting on the metro (instead of walking from the Marais). 

I also loved my time in the 8eme, around Bastille and Saint-Sébastien - Froissart metros. 

If you love views, an apartment up in Montmartre is your best bet. 

Take a look at apartment offered by VacationinParis. They are based in NJ,pay in US dollars, get key before you leave. My neighbor and I have rented from them several times and in several arrondisments. 

A number of people have already mentioned Quartier Du Marais, which is an excellent suggestion. Just a bit further east (east of Bastille) are both the 11th and 12th. Fewer tourists — not quite as charming as either the 3rd or 4th but it feels like a real neighborhood — with some of the best restaurants on Rue de Charonne and, from there, down toward Rue de Montreuil. 

I would recommend either Montparnasse or the Quartier Du Marais. Great 'real Paris' feel with minimal tourists and great walkable spaces.

In my opinion, the best are will always be the Quartier Du Marais, it's center to everything and just a really nice place. You can also look in the Saint-Germain-des-Prés (6th arr).

All good suggestions given.  As Christopher Frame mentioned, the 11th arrondissement of Paris & 12th arrondissement might give you a more "real" and "hip" area to base yourselves.

I took my 24-year old son to Paris and we stayed at an Air Bnb on Rue du Oberkampf. It was very diverse, had a lot of great restaurants (ethnic) nearby.  We had a great, large apartment with a secure access.  Hope this helps!  

Hi Rosalie,

Last time I visited Paris we rented an apartment in the Opéra region which was very close to the Palais-Royal and The Louvre It was walkable to a lot of places and I always felt safe, even late at night. I also love the Quartier Du Marais