I would say "neither", go for something like the Nikon D40x or Sony A200. Then, later, *if* you feel you're ready to get more serious you can leap the D60/A350 and go for something like the Nikon D90.
However, if in these credit crunch times you're lucky to have the budget (don't forget extra lenses, batteries etc etc all add up) for what you've asked, there's little in it. If pressed, I would say the Nikon D60.
The A350 and Nikon D60 are completely different specifications!
You should be comparing the Nikon D60 to the Sony A200, which is MUCH cheaper and still has a better specification than the D60.
The most obvious differences between the A350 and D60 are that the A350 has liveview and a higher mpix count. Then the Sony Alpha's have in-body image stabilisation, bracketing, an auto-focus motor.....
Only a biased Nikon user would dismiss a Sony and simply say buy a Nikon D60!
You've also to remember that this isn't a 'new kid on the block' thing with Sony. They bought the rights to the Minolta digital cameras so, these products are built on the Minolta provedence. All my cameras have been Minolta (not that I'm biased to Minolta, it's just that whenever I came to buy a new camera, Minolta came out tops at that particular time, in the research I did). The Sony A350 was a natural progression as I could still use all my orignal Minolta lenses. As for it being entry-level, well, take it with a pinch of salt you get a lot of bang for your buck with this, combined with the Carl Zeiss Sony lenses, I do a lot of pro work with this camera. Many folks have their preferred brand and many think by buying the best camera they'll take wonderful pictures . . . what works best for one, doesn't always suit all. There's a lot more to taking good pictures than having a good camera. Read the reviews and then go to a store and try the feel of the camera, etc you'll be surprised at the difference it will make.