Clicking on synaptic (which runs synaptic-pkexec), I can enter my password in the "sudo" (gksu?) window, but then synaptic GUI doesn't actually appear. I'm on Debian buster and LXDE.
Synaptic runs fine from the terminal (via sudo synaptic)!
Turns out I needed to install the policykit-1-gnome package.
It's the same issue as described in pkexec won't launch polkit GUI in Lubuntu / LXDE but on Debian. I have no idea why policykit-1-gnome isn't marked as a dependency for LXDE but it is.