My turn to be an obnoxious couple!
Foreign women with Korean men is quite a rare combination so I was expecting an increase in staring, but actually the amount of looks we got was pretty normal to how I am usually stared at on a daily basis. However, once while shopping together, the shop worker serving us asked if we were dating and was pretty impressed that we were. Usually the general attitude to Korean men dating foreign women is kind of congratulatory, which in some ways is good, but you also have to be aware of the darker side of that.
As a foreign woman you are kind of like an exotic designer handbag, there will most certainly be those who see you as you are and love you for it, and there will be those who just want to show you off to impress their friends.
"Sweet man! You collected a foreigner!"
Also if you are a foreign man dating a Korean women, you are more likely to run into violence than if it was the other way round. I've had male friends be beat up and threatened because they were seen flirting with Korean women. It's that whole "you can't come here and steal our women but we will happily steal yours" kind of attitude.
BUT! don't let that scare you. Most people are fine, you just to have to be aware of those that aren't. You also have to realise that maybe (like me) where you came from you were the ethnic majority so didn't experience life as a minority until now, or been such a small minority. Korea is not unique in this kind of attitude, it is just a general human thing when it comes to dealing with minority groups.
Family reactions
The extended Korean family have been absolutely lovely to me, even my boyfriends grandma was welcoming and accepting of me. I am lucky that I got a positive reaction. Not all families are as open minded, and that applies to cultures the world over, regardless of what your dating mix may be.
I was lucky, but I have known other mixed couples who did have trouble. The couple in mind had been dating for a long time until suddenly the parents on one side decided that although they were okay with them dating, the prospect of marriage was out of the question. This of course was difficult for both of the people to take, but in the end they decided to stay together. It's made family ties slightly tense, but I always say that once the parents adjust to the idea and realise that fighting just isn't worth it, they get over it. Also once you start giving them grandchildren they can't say no! I have friends with mixed race backgrounds whose parents had similar issues when they were dating, but they pulled through and now have a happy mixed family. So, my advice, hang in there!