I was on the train and a young woman, who was a stranger to me, was sitting in the seat across the aisle from me. She was clearly in distress, and periodically would begin to cry. In such a case how should care or concern (kindness) be shown without seeming to be prying into their affairs? Would a gentle "are you alright?" be appropriate? Would offering a tissue be an option, or might that be too vague a gesture?
Are you alright?
is fine, but if you feel less confident and wish to couch your language a little more then:
I don't mean to intrude but I wanted to check you are ok and if there is anything I can do to help
Then make a suggested offer of help: a tissue, a glass of water, someone to talk to, call a family member.
I think it is less direct, as it doesn't ask a question that the person may feel they have to lie to, i.e. saying they are alright when they are obviously not. It gives them an 'out' along the lines of,
I'll be ok and no there is nothing. But thank you for offering.
I personally think that showing kindness and caring should outweigh impropriety. That we should show caring for others even if it feels socially awkward or there is a social pressure against it - so please go for it and show kindness for someone if it is genuine and deal with the risk of them feeling you are prying. The up-side is you could change someone's life.
Adding to the answer, I think trains offer a special case of stranger interaction. Many European trains have those six-person rooms, and Amtrak (the US) will seat you with a stranger for dinner. We're encouraged to talk to each other. I've had interesting talks with people on trains, I wouldn't want to ignore someone simply because they're having a tough time.
I've also found psychological differences between conversations on the train, and say, random street encounters. I don't know what it is, maybe being in between here and there, life seems like it's on hold until the train arrives. An hour-long conversion with someone she'll never see again might be appealing to her.
I agree that "are you alright?" is a good start. You're making it clear that you see something is up, but also give a good 'out' for her: "I'm alright, just sad". It could very well be that she's not alright, as in, you need to go get help.