"We recently have been told that we can no longer transfer things to our mainframe with ftp. We currently use MVS::jesftp to submit some jobs."

Are you asking about "transfer things to our mainframe", "submit some jobs" or some combination of parts or all of those two things?

MVS::JESFTP is over a decade old (most recent version: 09 Jul 2003) and is just a wrapper around Net::FTP. The source code is not very long (possibly only a screenful after removing POD, etc.): it might not be a huge effort to write your own version to suit your needs using, for instance, one of the Net::SFTP::* modules on CPAN.

If you only want to transfer data, you could probably just use a Net::SFTP::* module as is. I did this a few years ago (as part of a contract) using Net::SFTP::Foreign. That worked fine at the time and, to the best of my knowledge, is still in production. I'm not in a position to advise whether that's the current best choice.

If you want to submit/monitor/control JCL, that's probably feasible but I don't know how to do it securely. The last time I did this was in 199? and used just plain Net::FTP (the code was similiar to what I'm currently seeing in the MVS::JESFTP source). Another monk will probably have better information: my best suggestion for an initial place to research how to do this would be Net::OpenSSH.