Sway bars will help a lot for body roll, but they are quite the expensive solution.
That said, thicker sway bars tend to be THE way to just do it and forget it.
There are some other ways to do it, though.
If you know how to drive, (i.e. either keeping it on all four wheels and/or take a corner on two wheels and not roll over) and prefer to have your SUV oversteer rather than understeer, you can swap out your small diameter 91-94 rear swaybar for the much thicker diameter rear sway bar that came on the 95-2001's. You'll of course need the bushings, brackets, and other mounting hardware. Cost wise, this can be done for whatever you can get the bar and hardware for, probably under $50 at a salvage yard, or even better, get it from someone on the forum parting out their 95-98 for way less.
Energy Suspension makes a full polyurethane bushing kit, or you can buy their sway bar and end link bushings seperately. I'd suggest the full kit since this lets you replace just about every rubber bushing that deflects with cornering with a much firmer polyurethane one. The kit says it's only for the 4x4's but almost everything fits the same on the 2WDs.
Cost of the kit is ~$100-120 depending on where you get it.
The other BIG item for handling is shocks. New shocks make a world of difference in body roll, and performance or monotube shocks make an even larger difference. Monotube shocks like Bilstein and the KYB Gas-a-just are good choices. Most other shocks are a low-pressure twin tube design, which is still okay, and they tend to cost less. Keep in mind that while firmer shocks help handling, they do affect ride quality. If you want a more luxury-car like ride, you'll want middle-of-the road shocks rather than the performance variety. Expect to spend ~$200-400 on good shocks.
Good tires help, too. Cheap tires have flimsy sidewalls that will flex more under hard cornering, better tires have firm sidewalls that won't give as easily.
It's important to keep tires inflated properly for handling, though. Even a performance tire will have a soft sidewall if it's underinflated. Higher pressures (up to the limit on the sidewall, usually (35,44, or 50 psi) make for better handling.
The last item is lowering. You can get drop I-breams for the front, or just get lowering springs, and either blocks for the rear or just de-arch the stock leaf springs. This gives a large improvement since the center of gravity is lower, and the replacement coil springs usually have a higher spring rate.
My suggestion for a budget improvement would be the bushings, and new shocks if the ones on there are old. It's possible to add in the larger rear swaybar, but you'll have to get aftermarket axle mount bushings (or find a way to enlarge the hole of the Energy Suspension bushings).
On the top end, you can't beat a ride that's been lowered, along with the sway bars, bushings, AND slightly shorter performance shocks. The improvement with all this makes the handling incredibly impressive for an Explorer.
