I don't get what's up with this "clutchphobia" among riders from Western countries, but here in India, clutched braking is an integral part of safe riding in the almost "nightmarish" metropolitan traffic.
Using the clutch while braking actually puts you in more control as you're free to downshift or upshift into any gear without worrying about the attitude of the bike at the current moment. It's only when you release the clutch that the changes take effect, so you have a lot of time to correct your mistake before taking off!
This is especially useful when riding in heavy traffic where you're constantly riding on the borderline of 2nd and 1st gear. Too high revs and you keep lurching the bike, too low revs and you might stall the engine. So keeping the clutch in is the best way to go.
Coming to engine braking, you don't even need that. Brakes are already way over qualified for their purpose so there's no reason you shouldn't be using them!
And finally, to those who say that holding the clutch in while braking would prevent me from accelerating in an emergency because I wouldn't be in the correct gear, who said I don't downshift when the clutch is in?
I regularly ride to work via a highway that allows me to do 70-80 speeds and then I have to take a turn, where I drop to almost 20-25. There I use the clutch and both my brakes to slow down, all the while also clicking down on the shifter according to my speed to be in the correct gear at all times. Then when I'm ready to takeoff, I simply let the clutch out slowly with steady throttle!
And let me tell you that it's smooth as butter, fast as a jet, safe as it can get, and demands way less attention than clutch feathering or throttle blipping.
And unlike those other techniques, this one won't F up your engine, tranny or clutch if you did it wrong!
TLDR: holding in the clutch while braking is perfectly ok (as long as you're downshifting with the speed). In fact, it's the correct way to brake, since your brakes are designed to slow you down, not the engine!