How do the characters in "A Clean Well-Lighted Place" reveal Hemingway's attitude toward our ethical responsibility toward others?
This is an awkward question to answer because it presupposes that "A Clean, Well-Lighted Place" does, in fact, reveal Hemingway's attitude toward our ethical responsibility toward others. There may not be textual grounds for analyzing the story as a disclosure of an attitude toward an ethical responsibility. This is because, textually, the story is analyzed as Hemingway's attempt to use the characters to reveal a means of avoiding nihilistic despair in the midst of existential meaninglessness. Ethical responsibility toward others in a world with no order and no meaning is not a philosophical discussion that is consistent with the philosophy of existential nihilism's meaninglessness and despair.
Having said this as a means of alerting you to the (seemingly groundless) presupposition contained within this question, there is a way to analyze character presentation to address the question of ethical responsibility toward others (but it does set aside the greater issue of creating meaning and order in an existential world thus does not tap into Hemingway's primary thematic issues). We'll examine this analytical option starting with the older waiter's comments and thoughts:
"I am of those who like to stay late at the cafe," the older waiter said.
"We are of two different kinds," the older waiter said. He was now dressed to go home. "... Each night I am reluctant to close up because there may be some one who needs the cafe. ... This is a clean and pleasant cafe. It is well lighted. The light is very good and also, now, there are shadows of the leaves." [...] It was the light of course but it is necessary that the place be clean and pleasant. You do not want music. Certainly you do not want music. Nor can you stand before a bar with dignity
In this quote, the older waiter, the one who appears to feel an ethical responsibility toward others, expresses his reasons for wanting to remain open an hour longer for the old deaf man. What he suggests here is that a clean, well-lighted place imposes a sense of order on a world that has no order of its own. It is well known that the world tends to a downward spiral, things tend toward entropy and deterioration. A clean, well-lighted place overrides the entropy by giving a momentary order. This waiter feels the absence of order and meaning himself ("he knew it all was nada y pues nada") and thus has empathy for others who feel it too.
This empathy can be interpreted as leading to a sense of ethical responsibility toward others who are suffering despair, like the old man ("He was in despair") and can be supposed to reflect Hemingway's attitude (though biographical/historical support would be needed to make a convincing case that this in fact Hemingway's attitude).
It might also be said that the young waiter is presented as a contrast to the older waiter to emphasize the importance of having a feeling of ethical responsibility toward others. in addition, the barman of the bodega who says, "Otro loco mas [You are very crazy]," to the older waiter presents an even stronger contrast to ethical responsibility because he not only feels no responsibility toward the older waiter, he outright insults him and in a similar manner to the young waiter's insult to the old man.