I can definitely recommend Benjamin's apartment for families traveling in Berlin. It has its quirks and could do with some serious renovation, as with most places in East Berlin I suppose (or at least from what my local friends tell me), but this is just a "don't let this freak you out too much when you arrive" message. Things work. Plenty of hot water in the showers (plus good water pressure), large and airy bedrooms with comfortable beds, big fridge that keeps your food properly cold, functional kitchen, Wi-Fi, small washing machine, proper heating for cold nights if necessary, etc. There is a baby high chair there, and from what I understand other baby stuff is available. The apartment is on the 2nd floor and there's an elevator, but you have to go up some steps to get to it.
The apartment is located in a quiet residential area, and it seems to be quite safe there. We saw a bunch of playgrounds here and there, but never got to use them due to the rainy weather. There are some small grocery stores closeby (Netto, Lidl). There are also an S-Bahn stop and two U-Bahn stops closeby (U8 and U5), each about 5 mins or so away, depending on how quickly (or not) you and the rest of your gang walk. Alexanderplatz is within walking distance. No restaurants nearby that we noticed (aside from a fast food place) but there are TONS of great places at Rosenthaler platz, a few quick stops away when you take the U8.
For those of you travellers interested in camping/outdoors, there's a great camping/outdoor goods/gear shop right by the U5 (Schillingsstrasse) stop :-D