Enabling LOAD DATA LOCAL INFILE in mysql

Question

LOAD DATA LOCAL INFILE is not enabled by default. Normally, it should be enabled by placing local-infile=1 in my.cnf. But it does not work for all installations. In my experience, it worked for Debian 7, but not Debian 7 minimal, though both installations come from the same precompiled deb package. Both on OpenVZ VPS.

How to debug why local-infile=1 does not work for an installation, and how to safely activate LOAD DATA LOCAL INFILE?