To start out use the search box above and search 'Paris apartments' and you will find tonnes of threads. I'll also attach a couple great threads if I can find them....

http://www.fodors.com/community/europe/paris-apartment-which-of-these-2-left-bank-apt-would-you-choose-and-why.cfm

What is your price range?

Try Hotel Minerve Or it's sister hotel right next door,la familia Excellent transportation options
or 5 bus lines across the street, good cafes and bistros nearby and bank and metro right up the street. Both have elevators. Rue des ecoles is within a short walk to the Seine

The Hotel Relais Bosquet has a couple of studios and an apartment which appears to be ground level. We love the hotel but have only seen the apartments on the web. The ground floor apt was already booked for our dates last year.
http://www.apartmentparishotel.com/appart42.html
We liked the idea of the space and privacy of the apartment and also having the services of the hotel.

I can't speak to bus service although there are busses in the area of course. We usually walk (we're in our very late 50s), take the Batobus and occasionally the Metro.

We ended up renting from Paris Perfect but would not rent the Kir again. It was a darling unit in the shadow of the Eiffel Tower but it was just too small. Our objection was having to move the coffee table to open the sofa bed and there wasn't a good place to put the table. PP was very good to work with so we would consider another rental from them.

Take a look at the Poppy or Marguerita apts from Guest Apertment Services. Both are ground floor and located on Isle St louis, so also quiet but close to lots, especially Notre Dame and Berthillon ice cream!

Virtually any hotel in Paris will have an elevator unless you look for 1* hotels. Even 2* hotels are required to have elevators if they have at least 5 floors, 3* hotels must have one if they have at least 4 floors (and that includes the ground floor). So if you want a hotel, just decide on your budget and area, it won't be difficult to find one with an elevator. Sometimes they can be real small, though.

As you're looking for an apartment, try www.homeaway.com, www.vacationinparis.com,
www.vrbo.com, www.parisbestlodge.com, www.airbnb.com.

One apartment that I had was a Vacation in Paris one #122 which had a nice elevator.

Happy Travels!

To help you map out buses for the things you want to visit in relation to other parts of Paris, see
http://www.ratp.fr/plan-interactif/cartebus.php

I'd suggest you also need to check that any apartment you're interested in is on the ground floor on an inner courtyard (so with a minimum of exposure to street noise).

I stayed in VIP #122 also,great area. Lately we have been renting their Marais Garden Delight. Ground floor with a courtyard . large bed and lots of storage space, everything you could possibly need across the street, laundomat next door, And several bus lines right outside the front door ,St Paul metro station 1/2 block away!