Does Safety Inspector kill himself if no other minions on board?

Yes, Safety Inspector can kill himself if he is the only unit you control.

According to the RiftJudge FAQ (ID 11053), when you play Safety Inspector, his "When you play me" ability is placed on the chain. If you did not pay the optional 3 XP additional cost, you must resolve the ability as written: "each player must kill one of their units." Since you are required to kill one of your units and Safety Inspector is on the battlefield, you can choose to kill him to satisfy this requirement.

If you did pay the 3 XP additional cost, the ability explicitly states, "If you paid my additional cost, you don't kill a unit this way," in which case you are exempt from killing a unit.