I recomend you Citadines Apartahotel for you and your family. It is not very expensive, me and my family stay once there, and my daugther stay there last year (october) with some friends and no problem.
This is the link, look for Barcelona.
The only thing is that this Aparthotel is in the Ramblas and you have to take care of your properties as kandajones says.
But don't worry and have a godd time in Barcelona.