The vmware logs were fairly useless so I looked around trying to get more information on the cause of the problem. This led to looking at /etc/rc.d/init.d/vmware. In order for that script to start vmnet it uses /usr/bin/vmware-networks --start and logs to the cleverly named /var/log/vnetlib. Looking at that log file, it's apparent that at least one problem is that the file /etc/vmware/networking is missing. There's no indication of why the installer failed to create it, or what it's even supposed to contain. However the following solved the problem:
# touch /etc/vmware/placeholder
# vmware-networks --migrate-network-settings /etc/vmware/placeholder
# vmware-networks --start
You can then just delete /etc/vmware/placeholder.
I suggest that VMware developers review and improve (a) diagnostic messages (the installer should have complained if it was unable to create or find a file that is absolutely necessary for a component like vmnet to run), (b) offer more detailed debugging (vmware-networks already has a --verbose option but it doesn't produce any information of value), and (c) unify logging so that it's all in one place instead of scattered around in various files like the poorly-named /var/log/vnetlib.