Wow. That provoked a lot of comments! Great! I love discussion.
There are a LOT of things that go into a good handling car. Many were mentioned above. The rear sway bar is just one part of the equation, but an important one as you get closer to "perfect". And once you address the other areas on a Corvette that need it, it becomes a more important part of the equation as you get the car closer and closer to that ideal handling balance. It's like the last tuning tool. I'll make a list here of my important handling considerations, based on Physics and my 27 years doing Pro-Solo and duking it out with champions.

In order:

• Modern high performance rubber- the wider the stickier the better
• Large wheels for the above since it is only available in 17 & 18" wheels today
• Keeping the tires flat in a turn - can mean a lot of things
• Good shocks to help smooth out body roll and bumps on the road/track so the tires stay planted
• Making the car roll less than stock - so transitions are quicker than stock
• Balancing the car front to rear so both front & rear work equally

That's really it but our C3s have 50 year old tech underneath. Some of it needs more help than others. A lot of this you really can do one piece at a time, but like an engine build, it's the final combo that counts. And the balance you want to strike between "easy-ride" or "kill" side of the handling equation. There are literally 100 combinations that "work well". they just have a different ride/handling compromise, or a different front/rear balance, or different cornering power.So I refuse to give you one perfect suspension "recipe". I know a stock steel suspension component C3 can handle 97% as good as a modern designed aluminum suspension component C6/7, but not quite. The basic geometry and parts weight are against it. But you can get really close even with stock parts. (Even without coil-overs.) But it'll be 1000% better than it was stone stock.

Areas of a C3 that need "help" due to 50 year old tech: (Not necessarily in order there is too much overlap)

• Tires - you just can't get sticky 15 " ones anymore. 17 is a reasonable minimum. What's the largest/widest tire that can fit in your fender before flares are needed? Flares just allow that number to grow. Flares are easier on the C3 than many cars. Wider is stickier and so is dedicated high performance rubber. You want both. But for geometry concerns you really want to stick pretty darn close to the original suspension designed 26" & something tire height, especially in the front.
• Wheels - all they do is support the tires. For handling a wheel should be as wide as possible, it shortens the transition time. Go with the max width the mfgr recommends for that tire. An 18" wheel also has an advantage over a 17" wheel because it allows shorter tire sidewalls and that means more cornering power and shorter transition times. A couple guys here run 10" wide wheels with stock fenders.
• The C3 has pretty good geometry even today (really good for a 1963 design) but it does have a lot of camber gain compared to most other cars. This is both good and bad but must be controlled. All the rest of the mechanical mods depend on how sticky of a tire you put on it.
• The back just has too much camber gain for today's rubber and the inner strut rod mounting point should be lowered aka a smart strut. That's actually a suspension adjustment area that most cars do not really possess. And even fewer know how to utilize.
• If you are going to jump from stock 0.8 G turns to 1.0-1.2 bushing deflection increases tremendously. At some level getting rid of all the rubber and going to poly/delrin/metal gets rid of the squirminess of the stock bushings and makes the car feel much more predictable. However I would start with the rear lower strut rod, and change that out, even on an almost stock car, and go straight to a metal heim joint there. Every other bushing can wait a while. That one carries 90% of the rear cornering force and makes the rear feel very squirmy.
• To me spring rates are easy. Better tires create more vertical movement. You need just enough spring to keep the car off the bump stops when on the brakes. In the rear you need enough to keep the half-shafts from binding with high HP. Any more than that and your ride/handling trade-off goes steeply downhill for very little improvement. Stiffer springs help the handling a little bit, but they quickly kill the ride. Turn in response only increases slightly, but bump compliance dies quickly, and the ride suffers quickly. The sway bars are responsible for 2/3 of the body roll, why try to do that with springs? The Vette is not too badly sprung for a good handling street car. I would not go to F41 springs on the street.Those are stiff enough for most racing. But we have very few options in-between. ( I may see if I can get a 350-400 lb front spring custom made, or just go with the 480s or the 300s. Decisions!!) The composite springs seem like a great ride improvement in the back. But the flipside is they can be difficult to get the rear ride height where you want it.
• Sway bars stock are very soft and let the car roll a lot. That was suspension design 101 with skinny tall tires in the 60s! They also, unknowingly by most, put the tire into too much negative or positive camber and it loses it's grip. Fatter ones are necessary with stock tires let alone good tires. Bigger ones serve two functions. The car body rolls a lot less, greatly increasing lag time when you turn the wheel. It takes a set quicker. And two the tire stays more upright, and has more traction. This last one matters much more on a Corvette than many cars due to it's camber curve. For this reason adding the bigger gymkhana sway bar to the front, all by itself, which is three times as stiff as the stock one, should "technically" make the car "understeer" like a truck, but in actuality it doesn't. The front tires stay much more vertical and gain a lot of traction so it stays way more neutral than the "math" says it would. So no the "numbers" don't lie, it just takes a lot of numbers to see the whlole picture. And there is such a thing as too much sway bar. If you make the car so stiff (with stiff springs & bars) that it rolls very little, it will feel quick responding like a go-cart, but it will have terrible handling on bumps or in the wet. The 1-1/8 gymkhana bar is pretty seriously stiff, and a good choice for most, but there are larger ones available. Tread lightly here.
• Drop the nose an inch to the Duntov racing specs in the Chevy power book. He designed it to the work the best at that height. The rear needs to stay about where it is due to the half-shafts. Follow Duntov's advice here on ride height, even on a street car. It's hard to lower this car due to the geometry in the rear. It's easier to lower the car with shorter tires, it'll handle better in the long run.
• More Caster in the front. As much as you can get. Stock is 1-2 degrees because of low speed manuvering with the manual steering option. 4-5 is achievable and helps several ways. It helps with high speed stability and nervousness so I would recommend it even on a stock car. It can even be used on a MS car if you don't mind it being a "bear" to park. In my mind the main thing it does is increase feedback to the driver to the steering wheel on what the tires are doing. You are basically trying to push the car "uphill" 1/8" or so and you feel this "pressure" as feedback in the steering wheel. At high turning speeds/forces the tire starts to slowly lose traction. You are now at the max limit of tire adhesion. You will feel this "turning pressure" lessen in the steering wheel. Caster helps you feel it. That's how a race car driver can "hold it there" at the tires limit, and not go "over" with likely bad results. Fortunately this "decreasing tire traction fall-off" is gentle on good tires. Peak tire traction/cornering force is usually somewhere around 107%, just a gentle slide or drift, but oh-so-close to the edge. If you push a little past that you will get less cornering force, and you'll get more drift or slide out of that end of the car. But "oh-boy" you had better have the handling balanced front to rear. If you can provoke the drift smoothly whenever the driver thinks its necessary, and also control which end of the car is drifting, it helps a lot. But you have to be able to "feel-it" and this is where the Caster helps. Example: On any given turn I would strive to be at 107-110% traction in the front on the brakes during initial turn in, and as I released it the nose turns-in, then at 107-110% on all four in mid-corner in 4 wheel drift mode, and then 110-120% or more on the rear coming out so I could power-slide the rear around a foot or two and use the power to help turn the car into the next straight. This sort of over-the-edge driving makes you a winner on the autocross course. On a high speed track it can be used to generate a "fast-lap" when needed, but is very nerve-wracking and hard on the tires to continue for long. Some autocrossers are very fast on a road course for this reason, but also very hard on the equipment. Rick Mears set a lap record at Laguna Seca one day in a street car, first time he ever drove it, by a couple seconds. After being congratulated his comment was "There was about 7 more seconds left in it, but that would have meant seriously sliding the car around. And it wasn't may car, I didn't want to break it." You have got to be able to "feel " the tires to get anywhere near doing that.
• Now you have to re-do your alignment settings a lot after all that. It's all about -0.5 degrees camber at Max-G. Then toe-in as preferred.
• The rear sway bar comes in last as a tuning aid to help you get that elusive "perfect balance" in a corner. It's like balancing a see-saw. There is only one spot where it is perfectly balanced, and the closer you get to it, it feels like a knife-edge, and the car tips over very easily to the other side I raced with adjustable sway bars on both ends for a reason. They were my easiest handling/driving adjustment and they have zero effect on all the other alignment specs in the car. I liked mine close enough to "dead neutral" that I could provoke the car into oversteer or understeer with either the wheel or the throttle. I called it my four-wheeled steering (It was an 80s thing!) Too close to dead-neutral and I had to be too gentle. A little farther away, (a little understeer) and I could be a lot more forceful with my driver inputs, especially the gas. After years of practice and driving schools I finally learned dead neutral felt great but was slow. The fastest way around the track was when the car was only "dead-neutral" when it was under full throttle power and in 4 wheel drift mode. Then I could power thru 75% of the turn, and be at that elusive 107% of tire traction by controlling the drift speed to the left. The longer I kept the car in "drift mode" the more the times dropped. Not to say I can drive like Rick Mears, but I kept trying.