Are you using an internal or external mic? Under the Input Source in the menu next to the record button you would need to select the correct input source. If it is an external mic make sure that the recording input is activated and not muted.
Otherwise, try deleting preferences and see if that cures the problem with the voiceover not recording.
To delete preferences open iMovie while holding down the Option and Commandkeys and select to delete preferences in the box that appears. iMovie will open in a new library. Reopen your old library to get back to your projects. Possibly you may need to reboot to set the changes. Deleting preferences is a safe procedure that will not cause data loss or disruption to your project.
If he above is unsuccessful, as a workaround you can use the Voice Memos app in your Mac's Applications folder to create a voice memo that can be dragged to your desktop from the Voice Memos sidebar. From there you can drag it into iMovie and use it as a voiceover. Not as convenient as the iMovie Voice Over feature, but still pretty easy, and at least you can finish your project.
-- Rich