Everyone makes valid points here, may I restate the concept better:
Brakes are cheaper than Drivelines (its more than the tranny, clutch, the axle, U-Joints and motor mounts are all taking a beating as well).
But Drivelines are cheaper than ENTIRE VEHICLE AND/OR YOUR LIFE
IMHO, Engine Braking is uneccessary for normal slow downs and stops and you should just use the brakes and avoid the shock and strain on the rest of the driveline.
BUT there are lots of abnormal stops and slowing that will overheat the brakes, and when they overheat they fade and/or fail. There are lots of accidents at the bottom of mountain roads from people that can't stop because they overheated their brakes riding them all the way down the mountain.
Understand how your brakes work, and look over your driveline while someone revs the engine or puts it in gear with the brake on to strain it, you'll see the stress's and you can figure out when its appropraite to just use the brakes or use engine braking to keep the vehicle at speed down extended hills/mountains.
Someone was asking if engine braking is bad for the motor? Perhaps, if the motor isn't particularly strong, riding down the mountain side while engine braking could put your engine under an extended off proper A/F ratio, the motor had to be pretty out of tune or poorly designed for it too bad enough to be anything but negligent. Mostly its everything in the power train BUT the motor, that takes the shock and strain, although not bad, its still adds up and adds to the wear.
Back in the CARB days, I had a friend thought he would save gas by turning the motor off as he engine braked all the way down the mountain side. Think about how a CARB works, all he did was turn off the ignition while the motor still turned, it continued to suck gas thru the motor BUT didn't burn it. At the bottom of the mountain, he turned the ignition on again and the entire bottom of his truck exploded in a fireball and blew the entire exhuast off the Pick-Up, leaving charred and burst pipes and mufflers on the road. The motor was fine, he just needed an entirely new exhaust.