I agree that Riverdale Avenue is okay in Yonkers close to the city line and Mount Saint Vincent, but then the area transitions to one that has issues. Shopping can be a bit of an issue in the immediate area, which is the same issue I would note with the third option. However, the third option is in a more stable neighborhood overall, and a middle-class area. The second option is middle class as well, as it is similar to the demographics of North Riverdale. Option three also has some small carryout restaurants and a tavern on McLean Avenue, and is close to Cross County Plaza for shopping with a large grocery store, mall stores, MicroCenter, etc.
With option one, it depends upon the type of basement apartment. Some are dreary, and others are decent, since there are many of the post-war townhouse style two and three family homes in that area that are built on slab foundations. You want to make sure that the apartment is legal, however, since there are some that are questionable. Check the city's BIS to see what the certificate shows:
http://a810-bisweb.nyc.gov/bisweb/bsqpm01.jsp Some have had the certificate of occupancy modified, since it was originally issued, but there are some buildings where the apartment was originally designed as an accessory to the owner's apartment, generally second floor, to preserve access to building systems, yard, etc. Since it is at a higher price, I would definitely make sure that the apartment is legal, since a converted accessory rec room can be a hassle with which to deal, needing to grant access to the landlord, tradespeople, etc.
If your niece lives in Westchester, she would not pay city resident income tax. Yonkers does have a 10% income tax surcharge for city residents, however, which is filed on a special form and is 10% of her state tax liability. So, if her state tax is $10k, for round numbers, Yonkers would collect an additional $1k. Since she would not be buying in the higher property tax areas of Westchester, it may be beneficial from a tax standpoint not to be a city resident. The other variable that I would check is car insurance because there is a surcharge for the city, but some areas in Westchester have lower levels of loss, and generally lower rates than the same driver would get in the city. The Bronx is not as high as Brooklyn in terms of car insurance, but higher than Manhattan and Westchester.
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