In many ways, I think the role of a senior student is to do and not to talk. By being a silent role model, other students will change more from influence, much of it internal, rather than external direction. This, I believe, makes for truer growth.
I think there are a few notable exceptions to this rule.
Before class: some people would say it's the senior student's responsibility to make sure the room is ready. Certainly, I like to show up at least 15 minutes early in case there are new students. Also, I think the senior student can help nudge people into getting into a "karate class mindset": sure, socializing is great (and for many, an important part of coming to class), but warming up can be done simultaneously. I think it's a good idea for the senior student to "check in" with people, to gauge their physical/mental/emotional state at that time.
Speaking of mental/emotional states, the senior student can be the most influential person for someone testing. Prior to class beginning, they can help the student rechannel their nervous energy into a more quiescent and focused energy. Because of the formality of class, this is a rare instance where the class fractures along an "us vs them" line.
After class, the senior student is often in an excellent position to followup a question or make a quick remark to another student. That said, I find it rare that I ever make unsolicited remarks to other students, but when I do, it's often immediately after class.
Lastly, while not solely the prerogative of the senior students, I believe that it is a good exercise for them to be writers, contributing essays that address contemporary issues.