I think I wouldn't blanket allow them or disallow them, but if I were a principal of a school K-12, I'd set clear boundaries:
- Adults/staff may have and carry any knife that is legal per state and local law. However they should have control over it (carried on their person or nearby) and not simply left laying around or in unlocked drawers.
- Knives for specific class purposes are kept in that classroom. Knives provided by the school of course are preferred, but I'd allow students to bring their own so long as they are turned in to the teacher when they arrive at school. The teacher would keep them locked up until the student is leaving the campus and wants it back, either when they leave for the day or at the end of the semester. I'd also allow a parent to handle the drop-off and reclaiming at any time of the day.
- Students who voluntarily turn in knives they brought, without any sign they intended something unlawful, are never to be punished. This includes students who accidentally bring knives in and do the right thing. They are not the type of student I should be worried about, and their honesty should be respected. They can claim the knife when they leave or it can be released to a parent.
- Students that are caught with knives outside of classroom use and having failed to turn them in, I'd treat it based on the circumstances. If it looks like the student just forgot about it and another student just noticed it and ratted on them, I'd just confiscate it and release it later to a parent, probably with a chide to more careful. If I caught them showing it to people or using it for something non-class related, this means they intentionally ignored the rules, I'd probably just give detention or similar. I have to do something because they need to have a consequence, or there's no reason to turn in the knife like above.
- Naturally, any student using a knife to perpetrate vandalism, as a threat, or any act of violence would be treated harshly according to whatever rules apply to property destruction or assault. Though I should add if the knife was used as a threat to prevent assault by other students, I'd do a pretty thorough investigation. I don't take bullying lightly and fully believe in self-defense, though the threatening student would still face some consequences for having obviously brought a knife ahead of time and thinking this was a good method of conflict resolution.