Barcelona is a great destination to take kids. It is very stroller friendly, from sidewalks to mass transportation, you can easily take your kids everywhere!
Having a car is totally unnecessary if you are planning to stay in the city. For your airport ride, cabs are allowed to take babies and toddlers in the back seat, without a child seat.
Having a small child with you is actually very useful in public areas such as crowded airports, museums, etc. Adults with small children can enjoy faster waiting time. You can skip most lines, including airport security and airline boarding crowds. Seriously, it can save you hours in the high season!
Just a fair warning, like in most European countries, you won't find convenient diaper change stations in every bathroom. This is when a portable changer is a huge help!