drag every time for me, (unless you read my recent post about loosing a big pike! lol) as i feel u can loose control when backwinding if your concentrating on the fish. i also find it can be alot more "jerky" than using the clutch, if im going after something big then i usually set my clutch on the slack side and cup the side of the spool when the fish runs to slow it down as i find this is the best of both worlds, it only takes 1 quick twist of the drag and tighten up enough to guide the fish over the net at the ned of the fight too. but theats just me 
Despite fishing for 35 years, i doubt i would know how to use the clutch on a reel. Always backwind for me....i'm totally opposite to Miggy above and feel i have far greater control by using the reel manually.
I suppose there's no right or wrong way, it's what your most comfortable with.
I favor backwind these days, after using the drag for years; I feel it gives me more control. If a fish wants to run, rather than 'wind' with my left hand, I'll just put my index finger of my right hand over the drum and hold or release it.
Drag for me - I ALWAYS used to use backwind when i first started fishing and ALWAYS used to forget to turn it off when i'd landed the fish, meaning the next bite is missed. Drags can be a bit unreliable however on reels in poor condition or very cheap models.
At the end of the day its personal preference I guess, but i quite like the sound of a good carp engaging the drag!
I used to backwind, but then I realised that I was probably giving fish too much credit on a run so I switched to using the drag, where every bit of line has to be fought for. This, coupled with a parabolic rod knocks the stuffing out of fish and sees fish in the net quicker.
I usually set my drag to be quite high so i know that if i do hear that unmistakable sound of the line peeling off the reel, i'm into a big fish. However, i usually use backwind if the fish cannot pull hard enough to take line as it's much easier to control the fish as i've had fish that take one last dive and take enough line to put me into a snag in reed or lillies.
I always used to use the drag (didn't it used to be called clutch?) but since starting up again I only use backwind, strange eh?
I also control it mainly with my finger on the drum, as well as actually backwinding.
I always use the drag (front clutch), or else whats the point in having it?
In my opinion, many anglers use the backwind because they do not use the clutch properly. They set it to just below the BS of the line which means that you can risk a hook pull if a big un suddenly tears off. I always set mine to a fairly light drive and lock the spool with my forefinger on the strike. I release immediately and this means that if you get a good un then you can give line straight away. Once on, control the pressure with you forefinger much as you would if playing a fish on a centre pin.
i have shimano reels for fishing the method, baggin wag and splasher (baitrunners) and use the drag at all times, super smooth and reliable,
for other types, normal feeder and waggler i have a couple of diawa's, and if i'm honest i'm not very impressed with the drag systems, notchy and sometimes just dont work ! smallest amount of much and they wont have it, so it's a backwind job !
but for my mind where big fish are concerned it has to be the drag.