My son and daughter-in-law chose an upstairs apartment so they could put a doggy door in the sliding glass patio door and let their little poodle have access to the balcony when they are gone to get fresh air and do her business on her pad which is kept in a holder out there. It works great for them and makes it pretty hard for anyone to get up there to take their little one. I can't remember what they used to block the bottom part off with, maybe lattice like the previous poster mentioned, but I know it was secure.
Another idea would be to use one of those play yards. They make one for human toddlers and pets but they are the same exact product, just costs more money for the pet one. I have one I used when mine were puppies. I will move my puppies that I have now from their little pack 'n play play pen to the play yard when they are another week or so older. Here is the play yard I have:
Amazon.com: North States Superyard XT Gate Play... but I got it on sale at Toys R Us. They also sell extension packs with two more panels in it so you could line the whole balcony if you wanted.