So.. not on public transport, then.
Other then offering to walk her back, anything else?
Well, the problem as I see it is that you want to ask her whether she wants to change your relationship, while ensuring that if she says no, your relationship remains fully intact.
In all honesty, what you want is not possible. Not just because she'll know you were interested in her, but because you'll know she knows that, and that she turned you down.
As for location, it's not as important as timing. You should probably ask her before you need to part ways for the day. Regardless of the answer, it'll be uncomfortable if she gives you her answer and then you just need to hang out for ages.
So really you'll need to do this in a way to allow you both to save face should she reject you, as she's free to do. So I suggest asking her to consider the idea of going out with you. Something like the following, although in your own words.
"Hey <girlname>, can I ask you a serious question?" - nobody ever says no to this. It also lets her know you're not just joking around.
"I want you to consider this a while before answering. We've known each other a long time, and I like to think we're good friends, but it occurred to me that it might be worthwhile seeing if we could be something more than that. I'd like you to think about whether you'd be interested in seeing where that might go."
Effectively, you're making her consider you as a suitor rather than just a friend.
Not entirely depending on her certainty, she will most likely answer right away, or will take you up on your offer of thinking about it.
Regardless of her answer, you need to have an answer prepared.
If she's interested, ask her if she'd like to go out on a date, and know where you'd want to take her.
If she isn't, suck it up and say something like "Well... it was just a thought." and play it off casually. If she needed time to think about it, you should thank her for considering it.