Well, I don't have a dojo (here sticks out tounge at those laughing at the very idea
) but this is one reason I am annoyed at those who carelessly bleed on others, the mat, etc., and train without covering wounds. Hepatitis is another reason.
I would make sure at LEAST that student knew s/he really did have to leave the mat immediately if cut, and I'd make sure everyone knew they HAD to wear gloves when cleaning up blood. The way to not make this obvious that you are worried about a student is to make this a routine practice (as it should be).
BLEEDING on the mat is much worse than peeing on the mat.
Same for your fellow students.
Try to avoid drawing blood--no jewelry, cut nails, etc.
Proper universal precautions for blood on the mat.
Good personal hygiene for all.
Maybe I am less paranoid than some might be, as I am a physician who treats patients all the time without knowing their HIV status.