I think that it is a difficult case to make that George is a reckless and irresponsible character. Steinbeck depicts him as the one who keeps an eye on Lennie and ends up taking care of both of them. There is little to indicate that he carries himself in a reckless manner. Rather, Steinbeck introduces him as a defined and focused character: "...small and quick, dark of face, with restless eyes and sharp strong features. Every part of him was defined: small, strong hands, slender arms, a thin and bony nose." Given this introduction, Steinbeck does not see George as a reckless character. Such an introduction establishes a focused and clearly defined element.
George is the responsible one. He is the one who secures the plan for his and Lennie's work on the ranch, develops "the escape route" for Lennie should he get into trouble, and is the one who constantly reminds Lennie of their shared dream. His responsibility extends towards Lennie in ensuring that the lynch mob won't harm him. It also carries through George having to shoot Lennie. The reckless and irresponsible action would be for George to let the lynch mob capture Lennie. George could not do that because of his sense of honor, responsibility, and love towards Lennie.