Avoid places that charge $1000/mo, haha, jk! It will be challenging to find a place that cheap in a decent area. I've seen $1000/mo for a studio and closer to $1200/mo for a 1 bedroom. I agree with Linda and CM, although I live in a nice neighborhood close to 87 (Willow Glen) but CM is 98% right. San Jose is very patchy, so it's hard to generalize. If you have an area in mind, you should repost and we can be more specific. Craigslist has a pretty comprehensive list of available rentals. I would search listings in Campbell, Cupertino, Los Gatos, Mountain View, San Jose South, San Jose West, Santa Clara, Saratoga, Sunnyvale, Willow Glen/Cambrian. That will give you a start to see what is available. Cambrian, Campbell and Blossom Valley are more affordable. When you do narrow down the search, make sure you drive by the place multiple times at night - I ruled out a place a few years ago that way even though I was born in San Jose. There was a stabbing there recently (!) San Jose is generally a very safe city with lots of nice neighborhood so don't be detered. Good luck!
ETA: Oops! I don't know how I missed the fixed income part. Here is the city website that lists low income housing options, including a section for seniors:
http://www.sjhousing.org/project/rental.html