Just trying to kick off some brainstorming for you...we live in a windy (desert) area, and I do find the wind really dries things out. Also it sounds like your plants do need more light. Any access to electricity, on your balcony?

Before we put up our little greenhouse, I built a plant shelf using this shelf unit from Sams:
Large 5 shelf unit on wheels

I purchased inexpensive fluorescent light fixtures, and hung them from the bottoms of the shelves. Added a timer to switch the lights on and off automatically. This is how I wintered my succulents inside the (dark) house during the cold months. Because the cart is pretty tall and heavy, maybe you could attach some kind of clear plastic sheets to the sides and/or back, to block some of the wind? Only thing that worries me about this idea is you'd have to find a way to safely shelter the electrical light fixtures from chance rain blowing in. (Water and electricity don't mix.)

Some apartment complexes might have restrictions against building or attaching things to your balcony railing, but I was thinking a free-standing cart on wheels might get around this (if it's a problem.)

You probably already know this, but the size and material of your planting containers, and the soil mix you use, also affects how quickly things dry out. For example, I need to water plants in small terra cotta pots a lot more often than plants in large plastic pots. My plants (succulents) need fast draining soil mixes, but you might experiment with different types of soil mixes that will retain moisture longer...depending on what you're growing.

There are also products you can add to your soil mix that will help hold the moisture in longer. Here in our desert climate, products like this are sometimes helpful:

Terra Sorb Water Saving Crystals

Hopefully others can chime in with some better ideas for you!