Your price range is reasonable for New York.
New York City is generally VERY VERY safe compared to other major cities. That said there are areas of North Brooklyn and South Bronx to be careful of. These places are not outright dangerous, but still require more caution than other areas.
Check the NYC Affordable Housing Resource Center at:
NYC Affordable Housing Resource Center
You should apply early before you even get to New York for Public Housing as the wait can be years, but Public housing here is Amazingly nice compared to other cities, and not just poor people. You qualify for incomes up to around $55,000 a year. Almost all developments have huge apartments, free electricity, heat, water, and have huge parks and trees all around. Search for NYCHA. All my assumptions of public housing were totally shattered when I got to NY. I think it is possibly the best that there is.
Section 8 on the other hand is good if you need it but I would not recommend, as buildings tend to be dirtier, more dangerous. And mostly very poor.
Try housing lotteries, There are many middle income apartment lotteries advertised in papers and that above NYC GOV website, search all those links, but those are the best bets, NYCHA, and middle income lotteries.
Being white, you will probably stick out in many neighborhoods, but you will be safe, and race isnt as big of an issue here since everyone is used to diversity. Taxi? No way, subway is totally safe, and Buses are VERY VERY VERY VERY VERY safe. Buses and trains are 24 hours a day and very extensive, if you feel more comfortable, you can get nearly anywhere by bus at night and they will stop where you want after hours to make sure that you are safe.