I think (C).
That every worker has a clean criminal record is of some importance to investment banks which is why a stringent background check is a necessary prerequisite for all of their job applicants.
1. That every worker has a clean criminal record is of some importance to investment banks which is why a stringent background check is a
necessary prerequisite for all of their job applicants
--necessary prerequisite is redundant
2.Clean criminal records of their employees is important to investment banks; hence,
a stringent background check are necessary prerequisites for employment
--a stringent background check is necessary
3.Because they consider it important that all of their employees have a clean criminal record, investment banks require each job applicant to undergo a stringent background check
--correct.
4.It is of some importance that all investment banks’ workers have clean criminal records
which is why many of them undergo stringent background checks
--'which' does not have a clear referrent
5.The reason that investment banks require background checks of their applicants is because
they require clean criminal records of their employees
--'they' is ambiguous
banks is plural, so we need 'they'