Having had Nikon for years, I'm going to be biased - of course.
I don't think there is anything wrong with the Sony, I think a lot of people are pleased with the camera. Some claim the colors are better. Their marketing for their sensor is a little misleading though - in a technical way.
I really don't know that much about the D60 other than it is one of the four consumer Nikon digital slrs. I've heard many people happy with the D40 and the D80 - just haven't heard anybody say anything about the D60. I just don't know anyone who has one. Try Fredmiranda.com, nikonians.com, and kenrockwell.com for some info. and to see if any complaints. Also might try amazon.com for comments, but they are sometimes not left by experienced photographers and sometimes novices.
The Sony uses Nikon lenses I believe, so there should not be any problem with them. I think one comment I heard that there are less buttons and more menus, but I would imagine that would be with both cameras. Just takes a little longer to make changes.
The Nikon autofocus and exposure is usually superior. With a Nikon, you pay for the name a little, but sometimes it is worth it. Unless, you are really set on the D60 - I'd look into the D80. I think then the D80 would beat out the Sony - definitely.
I'd be concerned with battery life, type of battery, what programs you might need to get to process the files, the type of storage, "noise" in low light conditions, focusing and exposure in low light conditions assuming glass and mp and flash are similar.