I would recommend Asakusa - it's a nice, quiet and cheap region. And it's easy to get there, basically almost every Subway line drives there and it,s about an hour and 20 minutes from the airport. (Furthermore it has got a few nice touristic attractions like the beautiful Asakusa Kannon temple)
Pls don't try to stay as near as possible to the airport, because a.) the train/subway system is so well organized that you'll have no problems to get to any place you want to and b.) you really would miss a great opportunity to see more of that fantastic country! it's awesome! ;D
I did stay at one of these backpacker hostels, it was called Khaosan Asakusa Annex. Good place to stay! Nice place, friendly people, low prices and you meet a lot of people which can help you when you;ve got questions.
Khaosan Tokyo Guest House Hostel :: Original ::
And noo, no worries, Tokyo is definitely not a dangerous place to stay. I spend whole nights on Tokyos streets and nothing ever happend.