Following a procedure I'd already tested on my home computer, I upgraded VirtualBox like this:

• Checked that the existing version and the target version were both available in the Synaptic Package Manager.
• Shut down all VMs, took snapshots, and backed them up. This includes backing up the contents of ~/.VirtualBox, which is on the root drive (while the VMs are on the second drive).
• Using Synaptic, removed the old version, and then installed the new version.
• Started VBox and tested the VMs.