Unless you're willing to spend a ton of money, my vote is for Asics. I've had good look with Asics cross trainers and, honestly, I've tried other shoes in the price range (below $60) and the quality you get compared to everything else in the price range is absolutely nuts.
I run in Asics, as do most of my friends who run. New Balance and Saucony are also really good. I'd try on some of each and see which you like best. Everyone's different so It's really a personal decision.
Asics all the way! I have knee problems and they totally help, but can be expensive (the style I buy to is $149)
I would definitely do some research and go to a shoe specialty store (like Ladies Footlocker) They can tell you what type of foot you have (more flat footed, higher arches, etc) and thus what support system you may need in a shoe.