How should university employees address student breakups (and drama in general)?
In general I would say it isn't your responsibility as a professor/TA any more than it would be as a normal member of the public witnessing a potentially violent altercation.

If you supervise one of the students you could ask them, at another time, whether everything is okay, giving them them the opportunity to talk if they wish. Avoid getting too involved, you're there to assist the students with their studies and risking effecting the relationship - if your interference has embarrassed or upset the student they would be less likely to approach you for help with their studies (your primary purpose).