When I had this problem, the culprit was a family of meadow voles. The first thing you need to know when dealing with these:
The best way to deal with voles is some method of extermination.
I e, netting really doesn't work.
I've had success with traps, and because I live across from a meadow, I've had plenty of chances to use them. I am a little leery of baits/poisons, partly because of the local cat population, which may not relate to you, and partly because I had to pull up someones porch in order to remove some poisoned stench-bombs.
I usually use mouse traps, and have had excellent results. They like peanut buttered apples. I usually use at least six traps, and try to place them in obviously frequented places, where plants will not blow in the wind.
In response to:
I live on the second floor of an apartment not too far from downtown so I am even shocked that an animal like that could exist out here.
In that case, this is not voles. The only large critter that digs up and eats seeds two stories up are squirrels.
I am not certain the squirrels around here are that vicious as I am not sure squirrels like to dig around in the soil.
Um... squirrels love to dig around in the soil. They are always messing up my mulching jobs, and love to dig out tulips as well.
I did suspect a cat but I don't think cats are vegetarians cause the culprit seems to love going after my yams and no matter what I do, it continues to go for it as it destroys other things.
Cats don't dig for yams (but nice visual). These are not your problem.
I am against poison but I don't think traps will do good here given I have neighbors.
I have watched squirrels chew their way through netting, wire, wood siding, and various other barriers. Would you consider a small live trap (I use one door Havahart® traps with success)? These do not harm the squirrel, and you can release them into the wild (far away).
You can also use repellents. I've successfully used Hot Pepper Wax on garden plants, reapplying every week or so. My neighbor recently started using Shake Away fox urine powder, with very good results. Also, using more than one repellent can be useful, in case your particular squirrel doesn't mind one as much.
I just tried the hot pepper idea. I liquified about 30 harbenero peppers and put generous scoops of the liquid on my plants. I hope that deters the critters from wanting any of my plants or seeds. I also hope it does not kill my seeds/plants.
I don't know how long the Capsaicin (hot chemical in peppers) will last in a homemade mix. You may have to reapply frequently. That is one plus for the hot pepper wax, it is has concentrated levels of capsaicin in a wax spray, so it lingers for quite a while.
If you put your mixture on in a thin layer, you will be fine, but don't put it on so thick the plant looks less green, because that will cut photosynthesis.
Also, put something on the soil as well, to prevent digging.