I'd like my NFS server to always use the same user ID to own the files, whatever user ID is used on the client.
Is this possible and how?
-mapall=user into your
man exports for details
This is possible, look at the User ID mapping section in the exports manpage.
Basically you want this in the exports line:
all_squash anonuid=whatever anongid=whatever
If you're not on linux but on BSD, do what Chris S said.