I suspect that over the course of one's career a PI becomes less involved with the personal lives of their grad students. In the beginning, one works fairly closely with the students in the lab because the lab is small and everyone must pitch in to get a new lab up and running. I think that this fosters a closer relationship between all lab members because the structure of the group less defined. Once the lab grows, a hierarchy has to be established and the PI (in most cases) spends less day-to-day time with the trainees and therefore gets to know them less personally than professionally.
However, as a new PI, I do know quite a bit about the personal lives of my students. I have spent time traveling with them, shared hotel rooms and many meals with them as well as countless hours in the lab and in my office, going over experiments and troubleshooting things. My students have been to my house for dinner with my family. In those circumstances, you get to know people. You find out about their lives, just as they find out about mine. I think the trick is figuring out the boundary between know about the personal lives of your students and being involved in said lives. It's a trickier distinction than it looks, as I have recently found out when one of my students had something of significance happen in their lives. I'm not the kind of person who tells someone to go to counseling services when they come into my office in tears, it's just not how I roll. At the same time, I'm only willing to advise on personal matters in a very limited sense - as in let someone know if they are doing something totally irrational, but that's about it.
So far the strategy I've taken is being supportive in ways I can, listen if the person needs to talk and encourage them to work on things that will serve as a good distraction but that aren't particularly demanding of a distracted mind. More established PIs may scoff at the notion of taking the time to address personal matters of grad students, but for me I would rather know there is an issue and help the person along than wonder why someone hasn't progressed in a month. Perhaps my view on this will change over time, but with a small group it seems like a good investment to ensure that personal set backs are addressed in a reasonable way so we can all move forward rather than have one person sink into an abyss and take their project with them.
4 hours ago