A blowout preventer (“BOP”) includes a housing comprising a vertical bore extending through the housing and ram cavities intersecting the bore and ram assemblies, each ram assembly movably positionable within a ram cavity. Each ram assembly includes a packer assembly that comprises an upper plate, a lower plate, an elastomeric body positioned between the upper plate and the lower plate and comprising elastomeric material, and a support member comprising an aperture and positioned between the upper plate and the lower plate to provide support to the ram assembly.