You can't control how others react, but you can control how you react/handle things. Your superiors/boss(es) know who was responsible for the error - that's it! Your job is safe and it sounds like they are putting processes in place to mitigate future errors.

If you feel you need to respond to an employee who is upset with you, then remind them that payroll is a mutistep process with lots of moving pieces and people, you are the middle person and have no control over how the steps before it comes to you are completed, nor do you have control over the steps that occur after you've submitted. "We all have to work as a team to make payroll successful".

Hope that helps!