The house is a great pick in Riomaggiore. It is close to the marina and right on the main street. You have a great roof terrace where you can eat and sunbathe (also a small other roof terrace, but we had no use for it). There is a tiny back yard (not well kept though, used mostly for garbage bins), but the kids had some space to play.
The apartment consists of one bed room, one living room/second bedroom, two bathrooms, one tiny space for storage (like a hall) by one bathroom and a small kitchen, plus 4 staircases (up to the first floor from the entrance door - up to the second floor - up to the big roof terrace - up to the small roof terrace). :-) It is clean and even the bed sofa is comfy. We got extra shower towels and separate beach towels, which was highly appreciated. You can wash laundry in a laundromat on the main street (4 € wash + 4 € tumble dry, takes about 80 minutes in all). You can buy washing powder (1€) or bring your own.
If you use 4 beds you have no sitting room (since the bed sofa is in the living room), which can be a problem if the weather is bad. Also the kitchen is tiny and if you are 4 persons around the table there is no room to walk around it. If you eat on the terrace this is no problem. The kitchen is pretty well equipped, except for wine glasses and serial bowls. There is eg. a Italian coffee brewer (used on the gas stove), electrical coffee machine, microwave oven, tee kettle and toaster.
I don't think you need to worry about the number of stairs. Our 4 year old did not complain, and after all if you come to Cinque Terre you are probably prepared to clime a bit.
All in all this is a great home base for exploring 5 terre. However, with kids I'd rather recommend Monterosso with a sand beach, promenade and playgrounds. There is a playground also in Riomaggiore on the hill. You reach it by first crossing the main street from the apartment, then take the upper hillside road towards the marina: great views! In Riomaggiore you can swim from the cliffs in the marina (even kids can manage it), the wave breaker, or at the "beach" past the boat dock with huge rocks (hard to walk on). In Manarola you also swim from the rocks in the marina. Vernazza has a small sand beach in its marina and a small beach facing the railroad towards Manarola, but we never went there (could be a rock beach).
PS! There is a great wine bar on the road reached through the back door of the apartment: FUORI ROTTA