Took hubbie and DD (2 yrs) with us and can thoroughly recommend it. We stayed in an apartment for a week and wanted to share the following points in case you're thinking of going:
We stayed in an area called Poblenou, to the East of the City. It worked v v well because a residential family orientated area. There is a long Rambla there (not THE Rambla) which is just full of families and buggies out strolling. Plenty of nursing mothers sitting on benches. Everyone made a huge fuss of our daughter which was lovely for all of us!
Restaurants generally don't have high chairs although they all have plenty of kids dining out. We took our Totseat with us which was fab and meant DD was safe and didn't keep getting down out of a 'big girl' seat while we were trying to eat. Restaurants did stack a couple of chairs on top of each other to raise her up to the right height, or put cushions under her, both of which helped.
We visited the Aquarium which was 17 euros each for adults for DD was free. Has an 80 underwater glass tunnel that is fab and DD loved it. Well worth the money to see her face and reaction to the fishes.
The 'magic fountains' are on the edge of Montjuic Park and start at 7pm every 1/2 hour Fri/Sat nights. Lovely display of lit water and music. Again DD loved it and was squealing with delight. Not too late for her either. The is Youtube coverage of it cos I've not done it justice here in my description. Was a great experience for us all and DD could just run around playing while we had a coffee and waited for it to start.
Pickpockets - we were so nervous about this as heard and read a lot about them before we left. Had no trouble but did see plenty of people doing all the things you are recommended not to! Leaving cameras, bags on/under tables etc. Just be careful and keep your belongings on you - the locals do!
Parks - very green city with lots of small parks and play areas everywhere. We visited lots and spent some very happy hours in Parc de Ciutadella which has several play/picnic areas.
As first time parents we discovered that you don't actually need to do very much with a toddler in a city in order to still have a really good time (all of you and not just them!).
Hope this info is useful if you're thinking of going, and that you have as much fun as we did.