I've mentioned this here before, about my friend/former co-worker Emma, who was in over her head at work and consequently very stressed about it. I had been assigned to help her with her project and we would sit in the conference room working, and she would get so stressed out that she would start sobbing, and telling me all the bad things in her life at the same time (sobbing so hard I couldn't understand her sometimes). (Which I understand is a bigger reaction than in the OP.)
Pretty early on I decided that I was just going to sit there quietly and wait for her to calm down on her own. I didn't actually do anything else (like read a magazine) but I thought about my grocery list and what was on TV that night and so forth. In other words I distanced myself from the situation and didn't take her distress on myself. Then when she had calmed down I jumped right back in to what we'd been talking about before. "Okay, so, I had suggested increasing the font size on these slides..."
I didn't want to escalate the situation, of course, but it also wasn't my job to be her counselor, and I found it unprofessional that she ended up crying at work so often, rather than exploring other coping mechanisms or doing other concrete things to help herself in life. So I just waited her out. I'm not a huggy, soothing kind of person, I'm afraid.
It's a little different because it was just the two of us in a room and I didn't have to worry about what third parties might see.
I think the OP said this doesn't happen often
, but it might be worth mentioning it to a boss and asking about any available training or policies. In some ways it's etiquette, but it other ways it's business policy--for example, if the OP has confidential material she needs to secure before leaving her desk or due to security concerns she can't leave a student attended in a room, that limits her options. Or maybe there are phrases or actions she needs to avoid using, for business liability reasons.