Well, I live near Moscow, so I don't really know where it's best to stay there.
But there's no actually dangerous districts of it as long as you don't do something amazingly stupid.
To save money I'd recommend staying somewhere on the outskirts - Izmaylovo hotel complex for example (three times as cheap as in the city centre).
The main factor here is to be near subway (Metro) station. Transportation is quite cheap (flat tariff of roughly one USD for trip for example) and you'll get to the city centre with Red Square and so on in 30 minutes.
Even if you are planning to visit World Cup, Izmaylovo is quite convenient, because you can get within hour to both venues that are in the city. To Luzhniki without any transfers and with one transfer - to Otkrytiye.