Some oil rigs accept passengers and mail, some accept nothing, and a rare few accept other cargoes.
Oil rigs should accept passengers and mail. If you use the Query tool on the squares next to the oil rig, one square will accept mail and another will accept passengers. However, half of the time this doesn't actually seem to be the case. Once in awhile an oil rig will decrease its production several times and then announce that it is closing. Even after the oil rig shuts down and disappears, part of it still remains.