Many Koreans have never eaten dog meat, and they'll usually cite one of the following three reasons for their avoidance of it:
1) They love dogs.
I think it's safe to say that this is why most of the Western world dry heaves at the very notion of eating dog meat. It's tough to conjure an appetite when you picture the cutest, cuddly (& very anthropomorphised) puppy you've ever seen, roasted on a stick.
This seems a weak argument to me, if the thought of eating meat doesn't already make you cringe. (The t-shirt's vegetarian campaign seems equally as likely to backfire.) Dogs are popular pets because they exhibit traits we look for in friends - they are loyal, affectionate, and can usually be trained not to poop on the couch.
Any animal can be made to look cute enough, if you take it out of its natural habitat and give it a bath and a ribbon, or if it is small enough.
2) It's cruel.
I only found out how the dogs are killed after my second trip to eat 보신탕 with a co-teacher. I had, perhaps naively, assumed that dogs were slaughtered as humanely as possible. But because the meat is especially prized for its stamina-inducing properties, butchers prefer to kill dogs at their most energetic, so that customers get the full effect of the dog's adrenaline. According to my co-teacher (on whose sole testimony this account is based, and who eats dog regularly), some butchers will torture a dog briefly before smashing its head on the ground. This supposedly kills the dog instantly, trapping the dog's adrenaline in its meat.
According to this 2001 article from the BBC, however, violent forms of dog execution have been banned, and they are now "instantly killed by electrocution." But, until late 2008, dogs weren't considered livestock (to do so would be to actually admit their "problem" to the world), so it has been difficult for the government to control their treatment. Now, having changed their designation, the government is trying to crack down on illicit dog meat vendors.
If you can stomach it, here is a photo gallery of some of the abuses suffered by dogs in markets in Korea and elsewhere: http://www.all-creatures.org/anex/dog.html. Keep in mind that many animals consumed in the US routinely undergo similarly cruel treatment.
3) They're Buddhist.
Korean Buddhists don't eat dog meat, and this accounts for almost a quarter of the population. While I had most often heard that Korean Buddhists consider dogs to be sentient beings (though dogs seem to be alone in the animal kingdom as deserving of that status), I was recently told of another, more entertaining and mythical story that started like this: "Korean Buddhists believe if they eat dog meat they will die." Dissatisfied with such an outlandish, cursory statement, I pressed for an explanation, which follows:
The kings during the Joseun dynasty, motivated by Confucian teachings, banished Buddhists from practicing their religion in the country. As a result, temples were moved high into remote hills where they were hard to locate and destroy. Practicing buddhists were forced to journey through still-wild, forests - at that time still home to giant Korean tigers - in order to reach their religious grounds. Tigers, apparently, love dog meat, and can smell it even on the sweat of a man who has recently consumed it. They would not hesitate to kill a man to sate their hunger for dog. Buddhists quickly learned to abstain from eating dog meat, lest they be killed by mistake.
Wikipedia will also tell you that dog meat has been illegal in Korea since 1984, which, if true, would probably do something to keep dog consumption down. Other articles suggest that it was the Seoul city metropolitan government that banned the sale of dog (and, incidentally, snake) meat, not the entire country. Regardless of the national government's "official" stance on the issue, personal experience tells me that the dog meat industry is wholly tolerated as a serious business. No attempt is made to hide the consumption from anyone other than foreigners. Most Korean fans of dog meat, upon hearing a foreigner's mention of it, will usually deride it blithely and pass it off as an unusual custom. Once you express any openness to the idea of eating dog meat, however, the same Korean will openly admit their passion for it and will invite you for supper, and probably lunch the next day. Ever since the '88 Olympics (and subsequently the 2002 World Cup, co-hosted by Korea and Japan), when the Korean government first suggested a willingness to shun dog meat as an official policy, the Korean dog-eating populace has gotten the hint: keep it on the down low, and you can keep beating - and eating - your (dog) meat.