In an outboard marine drive, an engine is supported on an engine mount case, and an under cover made of plastic material having a lower end attached to the engine mount case defines an open upper end which engages an open lower end of an engine cover also made of plastic material. A metallic stay member having a lower end fixedly attached to the engine mount case which is typically made of cast aluminum alloy and an upper end fixedly supporting a support rail reinforces the open upper end of the under cover. Thus, the essential part is reinforced by a metallic member, and the required mechanical strength can be achieved while minimizing the increase in weight. The upper end of the stay member may be used for attaching a latch unit for securing the engine cover to the under cover or the engine mount case.