Hi Jay,
Most importantly if Content Authors/other users are using the environment, warn them of possible down time. So, that no user of author instance is disappointment with the downtime. Possibly prepare a downtime message on Dispatcher to warn users.
You need to examine any comms that happen from Publisher to Author and from Author to Publisher.
Prepare correct steps to establish that there's no data loss with OOTB or customer data. Possible other steps that may be required due to customer changes made. Verify if this setup has stand by author.
Once all comms are paused, schedulers, workflows that may be affected are paused as well,
You can turn off/restart your Author instance.
Regards,
Peter