I'm a guy moving from NYC, so i can handle a bit.......
My wife is from Washington Heights, NYC, so I don't think you will find anything you see in SF scary. However, by California standards, some SF neighborhoods are not safe per se. But of course, Washington Heights makes Oakland or Hunter's Point or South Central look like paradise.
God forbid if the bad neighborhoods in San Diego, San Francisco, and Los Angeles become as run down as the tough neighborhoods in DC, Boston, and NYC. Every New Yorker and Bostonian, about a dozen in my family alone, and most in my wife's family, and many friends, all agree that nothing they have seen in the Bay Area and LA are that bad compared to east coast. Brother lives in DC and some say that's the worst on the east coast, if not the entire USA. The story about rats eating babies in NYC, that's largely a myth, but it has been recorded by a Scandinavian photojournalist in DC, with the Capital building in the background. At least the US Census Bureau looks at crime in every west coast neighborhood, but has long since stopped reporting the bad areas on the east coast, giving a casual viewer of statistics a very inaccurate view of just how bad the east coast crime is. East coast crime looks bad enough with what "is" reported, but if the census guys darned body armour and did go door to door in the South Bronx, then we could get some real stats, but I am not volunteering, mind you. Our murders out west get reported and we don't have rivers where bodies get discovered like y'all in the Big Rotten Apple. And our baseball stadiums have landfill under them, not missing mobsters!
It's when there is a homicide anywhere in California, and it's gang related, it makes page one that California city, sometimes statewide believe it or not.
In the West Coast NY Times, which we get even in small towns out west, a similar killing in NYC is nowhere near page 1 if it is not multiple and in the boroughs. I have to read some pages into the New York Times to see anything about homicides so that gives me some scale as to what is going on back where you are.
Anywhere in SF will both look cheap and safe if you have ever seen South Bronx, Brooklyn, etc.
As for SF being cheap, that's relative. Statistical Abstract of the United States of America puts SF at just over twice the national average in cost of living taking into account rent, food, transportation, clothing, etc.
NYC is 4.4 times the national average.