Does the Transportation Security Administration (TSA) offer some type of security awareness training for those involved in aviation?

Yes, the following link will provide information: Security Awareness 

Following is an exerpt from its introduction: Welcome to the Flight School Security Awareness (FSSA) Training Module! The purpose of FSSA training is to raise general security awareness levels of employees working in the flight training industry. Employees in this industry must be able to question and respond appropriately to any behaviors or activities that may present a threat to aviation security.

This training module will provide you with the basic knowledge on indicators of suspicious activity and how suspicious activity should be reported. It will also include a variety of scenarios to test your knowledge on what you have learned and will conclude with recommended best practices. Upon completion of the training you will receive a printable certificate.