Thanks for the vote of confidence!!!! This one is a hard one.
Left Luggage at the Estacio de Franca is probably the next best option if you can't leave it at the apartment. I'd call the apartment first.
We're doing the luggage in a rental car - but we're going all the way to Montserrat so should avoid concerns. Being able to park in a handicapped spot is also helpful since they are usually right up front in areas easily observed.
From what I've read on Barcelona crime, the biggest things to watch for are (and I went on a police website for this):
1) pickpockets. They are RAMPANT in Barcelona. Rick Steves also states that this is the most likely city in all of Europe where you will get pickpocketed.
2) Purse snatching - via scooter or otherwise
3) Someone in a scooter will slash a rental car tire to make you pull over. As you have pulled out all your luggage to get to the spare, a nice person will come to help you while their "friend" rides by in a scooter and steals your luggage.
Lesson: Avoid this by not looking like a tourist. You know how tourists look in your town - imitate the locals (e.g. no wide open maps). If you do get a flat tire, do NOT stop at the next most convenient pull-out spot (where they WILL be waiting for you). Instead continue driving until you get to a service station. This usually happens on the routes out of town, not in the city center.
There is less discussion of stealing luggage from trunks or even of car break-ins. Having suffered a car breakin before though, I suggest you leave everything out of site. Also, don't park the car, lock something in the trunk and then walk away. This gives people a clear view of that purse you just stuffed in the trunk.
HTH!
Sue