Latin Quarter or Le Marais

We love the Le Marais.

I have stayed in both areas , many times .

Really both are good, but trick is the specific street / location, as there are some very busy touristy streets in each, but also many quieter streets and areas.

I would include the 6th in your search also.

Read ads carefully , if it does not say building has an elevator, do not assume there is one for example.

Look at reviews , I would not choose one without reviews myself.

Also note crack down on illegal rentals , this is important. No one will ever admit their rental is not legal.

With kids that age, I'd stay over in the 6th Arrondissement - somewhere near Jardin du Luxembourg. Latin Quarter, while very convenient, is super touristy. Le Marais (also nice and convenient) is not that far behind it. If you really want to get out of the super touristy zones, you need to move into some of the double digit neighborhoods. Your trade off is that you will need to rely a bit more on public transit, taxis or Uber - things you might want to avoid with children in tow.

I'm with my Commonwealth cousin, Pat. Both areas are excellent with plenty of shops and restaurants. Some streets may be even too busy (noisy) for a family with young children, so keep that in mind when looking. The 6th near Luxembourg Gardens would be wonderful.

The 5th or 6th as close to the Luxembourg Gardens as possible (east, west or north sides) would be my target area if I had a 6 and 7 yo. Fabulous park that your kids will love!

One of my favorite places in Paris is the Luxembourg gardens! However, we found a wonderful apartment on a quiet pedestrian street in the Latin quarter where we had birds chirping in the trees outside our window and wonderful restaurants literary a few steps right around the corner. The street itself was very quiet ( Rue Rollin) It was perfect!

The vacation apartment situation in Paris will be changing soon, after the new laws that were passed in July 2016. Chances are good that an apartment you find now will not be available in April, unless the owner (not the agency) can obtain an official registration number issued by the City. You don't want to risk having to find last-minute lodgings for your family if the apartment you chose has been taken off the market.

The area around Luxembourg gardens is nice, but aside from that park, there isn't much to interest kids - so I don't think I'd necessarily recommend staying there.
Both the Marais and the Latin Quarter are heaving with tourists, but the Latin Quarter is probably the image that most people tend to think of when they imagine what Paris is like. The Marais is geared more to older teenagers and people from 20's - 30's - not much to do but look in boutique windows.

If you need a kitchen, you might have a look at this legal apart'hotel, which is conveniently located near the Cluny museum and close to transportation options, shops, restaurants, etc:
http://www.residencehenri4.com/en/
Directly across from the Louvre, right on the Seine, is another legal apart'hotel, which would be another good choice - same interesting neighborhood and good transport options:
http://www.citadines.com/en/france/paris/saint_germain_des_pres.html?utm_source=google&utm_medium=gmb&utm_campaign=CitadinesSaintGermaindesPr%C3%A9sParis
And in the same general area, here is a legal apart'hotel which is more budget friendly and still has everything you need:
http://www.thehotelnexus.com/hotel-near-landmark/Aparthotel%20Adagio%20Access%20Philippe%20Auguste/Latin%20Quarter%20-West%20Bank%20District/230121/131596

Aparthotel Adagio Access Paris Philippe Auguste is in the 20th arrondissement near Père Lachaise.