My favorite neighborhood in all of Paris is Ile St. Louis. We absolutely love it there. We spent a month in an apt there on our last trip, and have stayed there in a hotel before that. I've also stayed in many hotels, many times, in the 5th. Ile St. Louis is like a little village, surrounded by the river and beautiful views. Only the western end, along the main street gets crowded (in summer) with tourists from about noon-7 pm, the rest of the island is quiet and village-like. There's a really good grocery store, several bakeries, an ATM, a post office, pharmacies, produce store, butcher, flower shop, a church, an elementary school, a beautiful park at the eastern end, lots of fun little boutiques, many good restaurants and cafes, and great ice cream shops (Berthillon and Amarino).
It's a very short and easy walk from ISL to the 5th. From ISL/Ile de la Cite (where Notre Dame is) it's easy to catch a bus and there's easy metro access (Pont-Marie, Ile de la Cite, St. Michel).
There's a pedestrian bridge that goes from the back of Notre Dame to ISL, so easy walking, and lots of street entertainers perfom there day and night which is fun.
I don't know where the apt is on ISL that you're considering, but if it's not on the main street from the middle to the west end, it will be quiet.
I love the 5th, but prefer staying on ISL. The 5th, on the whole, is very busy, loud, crowded and lots of traffic.
Wherever you choose, it'll be fun. Enjoy!