My usual Mac user account is not an administrator of the Mac, Though some of the earlier suggestions in this thread helped me solve the failure, I had to modify the steps because of my non-Admin status. I post them in case they're useful to others.
In my case, I began logged into the Mac as non-Admin.
Stop the all VMs and quit VMWare Fusion ("VMWF")
Open the Applications folder in Finder, and drag VMWF to the Trash. Being non-Admin, I was prompted for Admin credentials, which I provided.
Open the Trash, rt-click VMWF in it, and choose Put back. Again, I provide Admin cred's.
Log into the Mac on an Admin account. Yes, I probably could have done that earlier, but I didn't; here I'm retracing my actual steps.
Then, as Mac Admin:
Launch VMWF. Since this Admin account had never run VMWF, I was prompted to create a first VM. I did not. Instead, I quit VMWF.
Open System Prefs > Security & Privacy > General, and click the button to unblock apps downloaded from VMWare, Inc.
In System Prefs >Security & Privacy > Privacy > Accessibility, confirm that VMWF is among the apps allowed to control your computer (my tick box was already duly checked)
After that, log out of the Mac Admin account, and go back to the usual, non-Admin account; in fact, I'd never logged out from it, I'd just switched to the Admin account temporarily, as macos allows.
With the non-Admin Mac account, launch VMWF and start a Windows VM. Plug a USB drive into the Mac, and (hopefully) say "Yay!" when you get the desired result (i.e. USB drive is accessible to the VM).
This is VMWF Professional v10.1.3 running on macos10.14
