I'm not sure what your problem is. What are you trying to do? Install a new version of Safari on your iPhone?? If so, the ONLY way to do that (stress "the only way", again) is update the iPhone to 3.1, which by the way, is a good idea anyway. I don't understand your reasoning for not wanting to update.
The reason you cannot get Safari 4 to install through iTunes, is that the ONLY version of Safari you can download on your PC is the version specifically for Windows, NOT for iPhone. Note-the only way to install applications on your iPhone is through the app store. Also, no it doesn't matter whether your phone is asleep or off when you sync.
Now, all that said, I have updated from 3.0 to 3.1, and Safari is exactly the same as far as I can tell.
Anyway, there isn't any reason to not update. 3.1 is more stable and a little faster than 3.0. It's a good update. Get it.
I spent so much time trying to figure out how to download/install only Safari 4 to 3GS from iTunes on my XP PC....finally called Apple to find out I cannot install Safari 4 without OS 3.1, and now I feel less stupid for not being able to figure it out. At this time, I do not want 3.1.
I see that my Apple Software Update program lists Safari 4 individually......would that be an option?....but then it would not go to the iphone.
That said, this might qualify as a "stupid" question -- Does it matter if the 3GS is on, asleep or off when downloading/installing?
Thanks for your patience with me