It's not easy - I also have a balcony with 3 others beneath. Frankly, I don't worry about it too much, I just water and let it run, but I am careful to water only in the evening, preferably just before dusk, when its unlikely anyone else will be on their balconies. I'm also careful with the amount - if the pot is very large and bone dry, I'll water with 2 litres, then come back half and hour later and give it more, so that water does not flood straight out of the bottom but actually gets retained in the compost. If its not bone dry, it'll get about 2.5 litres, but that's for a pot 2 feet deep and 12 inches wide containing a couple of climbers.
I do use trays underneath pots, particularly in summer when smaller pots dry out so quickly in hot sunshine, but I don't have anything beneath the very large pots or troughs. The bore with trays is you have to go back and empty them out after 30 minutes, and remember that they will collect water when the weather is cool and wet, and very few plants like standing in water. Standing pots on earth trays isn't a good idea - the roots of any plant in the pots will go through the bottom and root into the earth tray. Equally, standing them on something to raise them off the ground will not prevent water running out.