Actually, the cause would be safety if it were OSHA, and it would be as constitutional as prohibiting a worker from using alcohol (alcohol is often prohibited in amounts less than what creates intoxication). This would require that a substantial enough case could be made to keep a majority of voters from overturning it. OSHA does not now regulate a large portion of the workforce, so it would be a highly unlikely eventuality.
It would of course also violate the equal protection clause since people who were not employed would be exempted from the law.
That's just crazy talk. Like saying that DUI laws violate equal protection, because not everyone drives. Not every citizen has to be immediately subject to a law for it to be valid. Does the requirement for a pilot's license violate equal protection because not everyone is a pilot? Of course not. Your equal protection argument is beyond flimsy.