GALLIANO, Louisiana—On June 10, 1947, Stanolind Oil & Gas Co. won an auction for the right to drill for oil on a plot seven miles off the Louisiana coast. The company built a spindly steel platform and drilled a well in shallow waters. It struck oil, and in 1950, Stanolind sold its first Louisiana sweet crude for $2.67 a barrel.

More than 60 years later, the West Cameron 45-A platform is, according to government records, the oldest functioning platform in federal waters in the Gulf of Mexico. One of more than 100 structures...