Rue du Ranelagh-16th arrondissement

I understand your dilemma. As long as you're willing to stay farther out, would you consider other areas of Paris such as the 12-15th? Ranelagh although a lovely, quiet area of Paris is limited as to bus connections and metro connections. You might be spending more of your time on transport than you intended.

There is another option we are considering on the edge of Bastille and Marais. Would that be a better location?

Much better transport and proximity wise. Noisewise would depend on the location.

Should be fine. On one end at Ave du President Kennedy RER C station is showing today, Sunday afternoon in Paris, departures in the direction of St Michel/ Notre Dame every 15 minutes. On the other end, Ranelagh station on the Metro No 9 line, departures every 5 minutes to Right Bank of central Paris.

If you can give the nearest Metro station or cross streets to the apartment on the edge of Bastille and the Marais, it would be easier to give you valuable information.
Or, if you have the actual address, you could use GoogleMaps StreetView and investigate the neighborhood to see if you would be comfortable, there. Just make certain that you aren't over any bars, restaurants or situated on a noisy intersection.

It is on Boulevard Henry IV close to Bastille Metro.

IMO . . . very nice area. Away from the noise on the other side of Bastille but still within easy walking distance of most things I imagine you want to see. Good bus to St. Germain and Latin quarter. (#86)

rue de Ranelagh is a bit more peaceful and more upscale, while being a bit further out, in my opinion; Bastille/Marais/Henri IV "hipper" and closer to everything.

@ ctlambeth, I love the area of the 2nd apartment. I own an apartment on the same street and I find the area to be very convenient. I think it will be a great area for your sons to explore on their own. Rue du Ranelagh is very nice and I would probably live in that area if I lived in Paris full time but for a first trip, the Bastille location is much more convenient. Odds are that it will be noisier but it would depend on specific location and location of the bedrooms and whether the apartment's windows will need to be open at night, to determine just how quiet your space will be at night. For example, our place has the master bedroom facing Blvd Henri IV but our apartment has double glazed windows and we have US-style AC so it is very quiet. The back room looks out to the back of the neighboring building and is extremely quiet even with the windows open. Having quick access to Bastille metro station is a big plus and gets you easily to Metro line 1 for travel to the Louvre, Notre Dame, the Arc de Triomphe, etc. And you can walk over to Sully-Morland metro station in just about the same amount of time that it takes to get Bastille and take Line 7 to the Latin Quarter or Saint Germaine (with a change to line 10 at Jussieu). We have walked from our apartment to near Odeon metro station in 30 minutes and walked to Notre Dame in 20 minutes and to the Louvre in about 40 minutes, passing lots of great stuff along the way.

One other consideration for the Henri IV apartment is not only if the bedroom windows face the boulevard or the courtyard and if the panes are double or triple glazed, but also which floor. Ideally, it's a higher floor, bedroom windows are at least double and face the courtyard. However, people do survive on lower floors with windows on the street. And that would be a good neighborhood for convenience.

Bets has a good point about upper floors. Our apartment is on the 7th floor (American) and that really helps with the noise. Of course, if we had not put in AC, the space would be unbearably hot for my husband in the summer, which is when we are usually there. So one question is when are you traveling? In July or August, though I am rarely too warm, I would want to see if the apartment has AC.

Air conditioning will be a must, if you are travelling in July and August, especially if the apartment faces the boulevard - you'll have SouthWest exposure..
Boulevard Henri IV is noisy, there's a lot of bus and truck traffic, so it will be difficult to leave your windows open.

How close is it to Bastille Metro? The reason I ask is because there are often "manifestations" (protest marches, rallies) which begin or end up at Bastille.
Bastille Metro and bus lines will be closed for security, but you can walk in another direction to find another station.

If the apartment is in the rear of the building, you might still have noise problems if restaurants or cafes put their garbage in the courtyard.

I'd check out this apartment thoroughly and ask questions or ask for more photos from the owner, if you want to sleep at night.

The apartment at Ranelagh would not be my first choice.