Homestaykorea and Korea Homestay are both legit sites that I would look into. Although if you're on a budget at all, I'd sugest a goshiwon (basically the smallest apartment ever) or a guest house (usually a shared space). Goshiwons can seem kinda sketchy but are totally legit! My best friend in Korea was staying in one for about $300 per month with no deposit. It felt totally safe too. She and I always hung out in the room.