I have been to Barcelona four times and Girona once - but without children (bliss !)
For Barcelona I would consider staying in the beach areas (port olympic and barceloneta) rather than centrally. I have always stayed as close to the Barri Gotic./El born areas as possible as its great for shops, nightlife etc....but I presume you would want to be somewhere a bit quieter and closer to the beach?
The only thing to consider is the transport into the main bit - as I seem to recall that the metro doesnt run out as far as the Port Olympic. You could also try the Eixample - It doesnt have the nice, twisty old streets but would also be more child friendly than the central areas.
Definately visit Parc Guell - kids will love it and nice open spaces.
Also worth considering would be staying outside the city altogether and going in for day trips (I only have 1 DS aged 3 so the idea of getting around Barcelona with 4 DC fills me with fear :-). If this appeals consider Sitges - beautiful beach, nice shops and restaurants - 20-30 mins to Barcelona on a train.
Girona is lovely - great city walls. I stayed there for 1 night and 2 days - not much to do other than walking around the walls and city but you could definately pass a day or 2 there