I stayed at Deniss' place for a couple of days during the Fringe. He made sure that we all felt okay (as we were several people staying at the same time for the festival, his flatmates being away in August). The place is close to the city centre and Waverly station, which makes it easy to reach. It is quiet and safe indeed.
You'll get cereal, tea and coffee for the morning which lets you save you a bit of money. There is also a supermarket nearby, so that you can get all the products you need to cook you something good. There are many restaurants in the area but they are slightly expensive.
I would therefore recommend you stay at Deniss' place if you're on a budget.