Nanook was asked this question once by a blog fan - I still have the answer we wrote to her:
The Air Force Master Blaster: for Pooka, who had Nanook to observe, this was never a big deal. For Nanook, it was the TERRIBLE MONSTER WHO WOULD EAT HIS HEAD IF HE LET IT NEAR HIM! As with any fear impulse on the puppy's part, there is only one way to handle it - Mother Knows Best. Be calm, be firm, and be resolute. Do NOT pet her and coax her and tell her "it's okay baby, it's okay," with lots of cooing and so on - any fussing will convince her that this is something to be concerned about - it's reinforcement. Instead, be cheerful and firm. Think to yourself, "we are doing this" in a happy, fun way. Put that energy out there.
Most importantly - she does not get the option to run away scared. She should not have that freedom, at this point. She is too young to know what is good for her, so you are going to teach her.
First, put her on a collar and leash. Make sure they are both tight enough that she can't squirm out of the collar, (without choking her, of course). Then take her outside to the metro air force master blaster with a pocket full of high value treats, (hot dog bits, cheese, etc.) Ignore her, but make a big deal over the blow dryer, keeping it shut off for now. Pet it, talk to it in a happy voice, bounce around it, play with it in a low key kind of way - try to make it seem like the most fascinating object on the entire earth. Don't look at Sunny at all while you are doing this, just hold onto her leash so that she stays near you, and put all of your attention on the blow dryer. If you can get your kids to play along, that would help. In short, you want her to see that her entire "pack" loves the blow dryer, and thinks it is very interesting.
She should, (and I would nearly guarantee you that she will), become interested in it almost immediately - even if she's had a few bad scares involving it. She'll probably creep near, sniffing. Encourage her in an upbeat way with praise and treats when she comes near it, but no need to overdo it because again, she'll pick up on any anxiety on your part and it will reinforce her fear. Continue that sort of training until she is fearless and positive and herself around the blow dryer while it is off. This might take a few sessions - be patient. It will be worth it! Make the blow dryer a significant part of her day's training NOW, while she's still small enough to manage, so that you don't have to try and do it later when she's huge and holding her nearby with the leash becomes impossible. Since she is a puppy, and very flexible mentally, I would imagine by the end of one day consisting of several sessions she should be comfortable and ready to move on to the next stage.
Then you're going to go back to square one. Take the concentrator nozzle off the end of the hose so that the air flow is as gentle as you can make it, weight the nozzle end down with a brick or a chair or something so that the hose doesn't whip around when you aren't holding it (because at first you are going to concentrate on holding HER and just walking around the blow dryer), and only turn on one of the engines if you have the model that has two switches. Then, again with her on leash and held very firmly by you, (make sure her collar is tight and she can't wriggle out of it), you are going to do exactly the same sort of thing with it on. Happy play and bouncing and interest to the blow dryer while it is on, ignore her fear but treats and praise if she is brave and comes to investigate it. She is learning from you what appropriate behavior is for her "pack" around this strange monster. You want her to believe that the blow dryer is the most amazing thing on the planet.
Once she appears to be feeling more comfortable with the blow dryer sound, comes the hard part. You are going to have to put Mommy Knows Best into play. Keeping a happy but firm outlook and output with her, you are going to take the nozzle of the blow dryer, on its lowest setting and without the concentrator on the nozzle, and you are going to apply the air flow, gently, to her behind. Don't bring it up any further than her tush at first. So her back legs, her tail, and her hips are okay. Ignore any squirming and freaking out on her part completely. With nearly all dog training - ignore completely the behaviors you don't like, and reward the behaviors you do. Mommy Knows Best, and she's just going to have to get used to it. If necessary, stand on her leash, or tie her to something to keep her semi-still. Do not respond, at all, in any way, to fear on her part but reward good behavior - bravery and standing still - with treats and praise. And if you start feeling guilty for 'torturing her,' just keep reminding yourself that this blow dryer is an important tool in your arsenal to keep her healthy - it will help prevent skin issues when she gets wet, and will remove dander and dead fur - and so it is necessary to her and to you and she's just going to have to suck it up. *grin*
With Nanook, I tied his leash to a gigantic metal outdoor bench, stood on the center of the leash with one foot to keep him near enough for me to play the nozzle flow over his tush, and held a handful of hot dog bits in a closed fist to his nose with my free hand. When he stopped trying to run away and started instead to try and nuzzle his way into my fist for the treats, (which would make him stand still), I would praise him and give him one of the bits. At first I would also turn the dryer off and that would be the end of that session, but I gradually increased the amount of time I expected him to stand still, so by the fourth session he'd get two or three bits before I turned off the dryer...etc.
Keep sessions short at first, and try and end on a good note - if she stands still for you to dry her, even for a few seconds, immediately praise, treat, and shut it off and go play a quick game of whatever she likes best as an extra reward. But always make it clear that the session ends when YOU decide it does, and not when she does. Like with everything else, you'll gradually increase the amount of time you do it as she gets accustomed to it.
Once she stops really reacting in an extreme manner to the sound of the blow dryer, and the feel of it on her tush, you can start to move it up her back and do around her chest and neck, with treats and praise for good behavior as above. I usually towel dry the head and ears, but my guys both LOVE to have their chins and the underside of their necks dried because they are always so itchy there! They fight over who gets blow dried first by shoving their necks into the nozzle when I turn it on!