Dalmatian

Just seconding that I don't think it's a "responsibility" but instead, just something that is the right thing to do professionally! Like AP and fuzzy, I also made the effort to reach out to new students every year. The two things I try to keep in mind were: 1) try to reach out to everyone equally, not just a couple of "favourites" and 2) don't push it if the junior student doesn't seem like they are interested in it.
For 1) I worry that certain types of students get more mentoring than others because some studies show that mentors are more likely to spend more time with mentees that are similar to themselves and/or certain personality traits get "selected" more often when senior people choose mentees. So I try to make an extra effort to talk to all the new students approximately the same amount of time for the first semester each year. We get on average, 4 students per year, so this is not too hard. It's just a matter of thinking about how often I start a conversation with a student and ensuring I am not favouring one over the others.
For 2) I try to gauge the student's interest in further discussions. In the first semester, during the "talk to everyone" phase, I try to continue initiating conversation with new students, unless they are very clearly not interested in talking to me. After the first semester, I'll probably check in with them to see how they are doing during critical times (e.g. quals) in the first year in case.
It doesn't make sense for every senior grad student to "mentor" every single new student so of course, I don't continue engaging everyone at the same level after the first semester. By then, there are some students that where more interactions between us would be great for both and others who are better off just knowing the senior students exist if they need help. It's all informal so it's not like any one has a senior grad student mentor "assigned" to them. But on average, there is 1 student each year that I click really well with. For the others, I try to keep an eye out for them, especially if they are more quiet/reserved. Nothing wrong with that of course but I'll try to check in on a student if I notice a change in behaviour or if they stop showing up to events, classes etc.

I don't think it's a responsibility exactly, but I think it's the professional thing to do as a senior student toward junior students. Junior students will often take time to adjust, especially if they have some unusual circumstance (first generation, international, etc). Not everyone will reach out, and they won't always know who is a good person to talk to -- that takes time. It's just a nice thing to do to reach out to those students once you've been around for a while and help them out. It's often part of the department culture. If someone did that for you, you'd be more likely to do it for someone else. You don't even necessarily have to be senior -- even a second year student will have a lot of wisdom to impart on first-years. 

I think this is very specific within departments, programs, and fields. 
I am in a small department, and my field has one student per year so it is very easy for us to create those mentorship bonds. American historians are many more (maybe four or five per year) so their dynamic is completely different. On the one hand, not all senior students "feel" they need to talk to first years. On the other, because they are more on campus while the rest of us go out to the field, they inadvertently become de facto mentors because they are around. 
When I was in my first year, my senior grad students approached me, helped me find books, drove me places, invited me to gatherings, etc. I doubt I would have done it myself. As a senior student now, I approached first years in the same way. I think it is a good way of creating a friendly collegial working environment, especially since we already have too much going on as students. But many people in my department disagree with me and think the opposite: we are writing the dissertation, they should approach us. I simply think that if you want something, then do it. I want people to feel welcome, so I go and welcome them. 

I think it is a great moment for you to realize that you begin to adapt your new life. I am a first year of Ph.D. and my relentless mentor made me cry 3 times. I already went through a mental breakdown and actually yelled at my advisor during one of the lab meetings. Thank God everyday if you have a supporting faculty around you. I was a very successful undergrad and had an amazing master's experience, but I have lost all my passion and ambition in the Ph.D. thanks to my mentor. You will be fine but I won't. That's why I am doing my best to live this goddamn place. 
 
 
 

In undergraduate studies, I ended up crying an office maybe 2 times. Typically, this occurred with professors I'm very close with, and I don't want to disappoint. Each time the professors were very kind about it, and I was given the impression it wasn't entirely unusual. They often offered excellent support by relieving some of the burden of the situation. 
I graduate school, I ended up crying once in front of a professor. It actually really improved the relationship. The professor hadn't realized that the way he was communicating with me was coming across mean and dissatisfied, and we came to an understanding. 
I do find it embarrassing that I couldn't keep it totally professional, but I'm certain they recognize the stress we are under during graduate school (especially when we are first adjusting). 

I cried in a professor's office because of a medical issue (I was dealing with a really bad concussion) and I couldn't do exams or anything. It was mortifying and I was trying to keep it together, but he was very kind (and knew me from undergrad). It happens.

@Dalmatian Yes, I have had students cry in my office, usually in circumstances not unlike what you're describing. It happens. Not much you can do about an uncontrollable emotion that sweeps over you, so I'm glad to hear that your professor handled it well. 
@Berk Oh, that really doesn't sound healthy for someone in their first semester. I would strongly advise you to look for support elsewhere, be it through moving to another lab/supervisor or by simply finding a mentor who isn't your advisor. It can be an advanced student, a faculty member, maybe someone in a specialized support group (those exist!). But don't just keep going like this for too long, because you'll find it very hard to finish, and I'd personally question whether you should. A PhD is not worth 5 years of suffering.