Bathos, in literature, is the often unintended shift or lapse to the ridiculous or commonplace when attempting an exalted or sublime effect. In other words, it is when a poet or writer attempts to write with sublime style, intellectual content, and artistry and ends up writing sloppy or dumbed-down material.
In "Culture and Anarchy," Arnold was making a careful argument that anarchy, which he compared but not on totally equal terms with free will, is not necessarily the best way to go. He wrote that we (English at the time) favored free will for itself, not considering the possibility that there may be higher ideals or forms of culture that we might aspire to. He was careful not to argue for stronger centralized authority without reservation but his point was to propose that the free will doctrine to "do as one likes" (Chapter 2) is not necessarily the best simply because it entails free will. Bathos, in this context, means to treat any method or style or lifestyle as on equal ground. That is, those who favor bathos believe that any means of education, art, or social institution are sublime and intellectual (simply because it was chosen with free will). Thus, Arnold argues for, at least the acknowledgment of, the possibility that the means of achieving the goal of culture must be put under more scrutiny than saying "do whatever you want." He equated bathos with an "anything goes" or "anything can be sublime" stigma.
This argument is held today in political and artistic discussions. Postmodernism, in one of its many definitions, is often described as an "anything goes" doctrine. This approach does free up artists and citizens to live their lives how they wish. It is a freeing, liberal doctrine and should be praised as a liberating philosophy. But Arnold was arguing that, while freeing, this "taste for bathos" can dilute human potential. He wrote that the end goal of culture should be human perfection. Therefore, the means of getting there should liberate the individual and society as a whole, but we should establish some guidance and always seek the best standards for ourselves as time goes on. The "anything goes" anarchy where we exalt everyone's "taste for the bathos" in attempts to challenge authority and acquire free will is liberating but it is only aimless wandering unless we, at least to some degree, position ourselves toward improving the human condition.