On a quiet road you can first...
On a quiet road you can first practice just matching the engine rpm to road speed while downshifting. A clean downshift with properly matched engine and road speeds should hardly upset the chassis at all.
Even though many of today's sportbikes with slipper clutches let you make sloppy downshifts with little penalty, there are still situations where proper technique can make for a smoother corner entry, even with a slipper-clutch-equipped bike. The key is to correctly match the engine's revs to the bike's speed, which may seem obvious but is difficult to master in practice. In addition, there are situations where the process can be modified to other advantages
You've probably experienced the results of a ragged downshift. The rear tire snakes as you let the clutch out and the slow-spinning engine drags the rear wheel speed down suddenly. The key is to blip the throttle when the clutch is pulled in, raising the engine's speed so that the rpm is correct for the gear and wheel speed when you let the clutch out again. For example, if you're cruising along at 50 mph and 5000 rpm, dropping a gear will raise the required engine speed to roughly 6500 rpm. Not blipping the throttle on the downshift will have the engine close to idle, whereas you should have it spinning close to the required rpm before letting the clutch out. It's a lot to accomplish in a short period of time, but with practice giving a short burst of throttle during a downshift will become quick and second nature.
To properly blip the throttle...
To properly blip the throttle while braking, you'll first need to get used to two-fingered braking as you'll be using the other two for working the twistgrip.
Rather than jump in the deep end and try everything at once, practice with your bike safely parked. It's practically impossible to work the throttle when you have all four fingers on the brake, so you should be accustomed to two-fingered braking before attempting this. With your bike parked and turned off, work the throttle while applying the brake to become accustomed to the required motion. Add the clutch to the act, pulling it in first, then blipping the throttle, then letting it out slowly. Now start the bike, and repeat the exercise with the engine running (er...and in neutral, obviously) to see the relationship between the throttle and engine rpm.
Ready to ride? Find a quiet, straight stretch of road and at a midrange rpm first try simply dropping a gear and blipping the throttle to match the revs when the clutch is let out. If the engine rpm is perfectly matched to road speed you'll be able to let the clutch out quickly and not feel the engine drag the motorcycle's speed down much if at all. On the vast majority of sportbikes, it shouldn't take much throttle movement at all to raise the rpm the required amount; a mere minor flick of the wrist is usually all that's necessary. Once you can make a smooth downshift, bring the brakes into the equation, lightly applying them while downshifting. Experiment with varying amounts of braking and engine speed to find what's required with different combinations. Ideally you will want to be able to smoothly downshift and blip the throttle using high rpm and hard braking, and at that point it would be safer to experiment on the track rather than even an isolated road.
The goal you want to work...
The goal you want to work toward is downshifting at maximum revs and hard braking while slowing for a turn at the racetrack. The better you are at downshifting the smoother your braking will be, allowing you to brake harder and deeper into the corner.
With a bike equipped with a slipper clutch, an alternative is to not blip the throttle but rather slowly let the clutch out between downshifts and use that to help with braking. This is especially useful on the final downshift, as it can allow you to release the front brake earlier than otherwise, taking some load off the front end at a critical point. Before you go blazing into a corner at the track and attempt this technique, however, note that it requires quite a bit more concentration to accomplish than just blipping the throttle. You should first try it in a straight line to see how the chassis is affected and how much smoother you have to be in letting the clutch out to keep things under control. In an extreme case, racers will use the clutch to step the rear end out during hard braking, setting the bike up in advance for the corner.
In a decreasing radius turn, you may have to make the last downshift as you are trail braking into the corner-your speed at the beginning of the corner is too fast for the gear required for the tighter, final portion. Often riders will sacrifice the exit of the corner to avoid the downshift entering the turn, leaving the bike in a gear that is too tall for the exit. If you are smooth, however, you won't be upsetting the chassis under braking and will be able to confidently make the downshift while trail braking. Although downshifting while leaned over may seem the only option for a decreasing radius corner, you can avoid that scenario by taking things a step further. While you are still upright and braking for the corner, make the final downshift. If you were to let the clutch out completely at that point, however, the engine speed would be way past redline. The trick is to blip the throttle so the engine revs rise to what you'll need later in the corner, then very gradually let the clutch out, so that the engine rpm stays at that level as you are braking into the turn. This calls for some very precise clutch work and will take some practice to get right, but lets you skip a downshift while trail braking.
Being smooth on the final...
Being smooth on the final downshift into a decreasing radius corner is where this technique pays dividends. The more precise you are with the downshift, the smoother your braking will be and you'll find that you can brake later and deeper into the turn.
Even though you may not have to blip the throttle on every downshift, it's a good habit to get into as the more you practice the smoother you will be and the quicker you will be able to make each shift. Eventually it will become second nature and you will have perfectly matched downshifts without even having to think about it.