What many foreigners do, who come to study at Yonsei University's Korean language program, is stay at a Hasook-Jib [하숙집], basically a private boarding house where you get your own bedroom, with a desk, bed, and internet access. The person who runs the boarding house generally cooks 2-3 meals a day, so you do not have to necessarily go out to eat, but can of course if you feel like it. Sometimes the owner will do your laundry for you, but I'm not too sure, it depends which one you stay at. Of course all utilities are included, but the downside is that the bathroom facilities are almost always shared. Usually the min. price of win is between 200,000-300,000 WON a month, but I would recommend the more expensive ones if you can afford it, somewhere between 400,000-600,000 WON a month, as they have more updated facilities are generally much cleaner. Sinchon is regarded as one of the more expensive districts in Seoul because of the high demand for space, the lack of building space, and the 3 large universities located there (Sogang University,Ehwa Women's University, and of course Yonsei University). Hasook-Jib's almost never have a deposit, but some may have a deposit of a few hundred thousand WON, though that is rare. I have many friends who are staying at Hasook-Jibs in Sinchon, and if you want, I can ask around to help you find a good one, though your best bet might be to e-mail Yonsei University's Korean Language Institute (KLI), and ask for a list of Hasook-Jibs, and they will send you a list of Hasook-Jibs near Yonsei with the approximate prices per room and the telephone number.
There are also what are known as Goshiwons [고시원], which are considerably smaller than Hasook-Jibs, but the main 2 differences are that usually you have your own bathroom in the room, though the rooms are very tiny, and what separates the room from the bedroom part is simply a glass bathroom door,so basically you are sleeping about a meter or two away from your toilet, shower, and sink. The second difference is that no meals are provided, but usually Kimchi, Rice, and Ramyeun are there for the taking in a shared kitchen. Also, there is a washing machine in the kitchen to be used whenever. But people chose Goshiwon's for their privacy, as you have your own bathing facilities. The only Goshiwon in Sinchon that I know of that is decent, is Soho Livingtel. The website is plagued by spam and spy-ware, so do not open it, but let me know if you would like more information.
If you do a Google search for the Hasook-Jib or Goshiwon, there are sure to be numerous results, as there are any foreigners in Sinchon. I hope that helped, and of course you can contact me any time, as I am not living in Sinchon right now, but can help if you need. My e-mail address is KyleKim560@Yahoo.com.
Also, if you search a Korean search engine such as Naver.com for 하숙집 or 고시원 you can do a search, though you will need to able to read and understand some Korean.
Yonsei University should be a big help though, at least the KLI should be. Also Dave's ESL Cafe has many forums in which you can find much Information on Yonsei, Sinchon, Goshiwons, and Hasook-Jibs.
Good luck, and I hope you enjoy your time in Korea!
P.S. Have you contacted Yonsei about the i-House dormitories (The International House)? The i-House 2 just opened about a month or two ago, and the facilities are top notch, very nice and clean. Worth checking out, if you can get into a room.
Been living in Seoul for the last 8 months or so.