This question presupposes the text pits the "brute within" against the "veneer of civilization." This will be a hard case to make and will hang on a very circumstantial thread.
The text makes it clear that these four men, all of different classes in life--an engine broiler operator, covered in oil and soot; a cook, spotlessly clean serving travelers; the captain, dignified and catering to traveling customers, not cargo; a news correspondent, using intellect to report about the world--are living in this life-and-death situation, exhausted to the bone, according to an unspoken though shared ethic, a shared civility and courtesy, a shared morality of human dignity and rights. Let's look at these three, then propose the one thread where "veneer" might clash with "brute."
Shared Ethic: After the cook expresses optimism over the on-shore wind (blowing toward shore), all are a bit shamed by the captain when he downheartedly suggests they have no hope even with the wind: "Do you think we've got much of a show, now, boys?" The narrator expressly states that the ethic that binds them prohibits the acknowledgement of hopelessness; it might crush the courage and spirit of the others. This passage directly contradicts the idea of a "veneer of civilization" and a "brute within" and disproves it:
the ethics of their condition was decidedly against any open suggestion of hopelessness. So they were silent.
Shared Civility and Courtesy: The men are in "the last stages of exhaustion." What they say and do is genuine and sincere since all veneer and posturing have long been washed away by stinging salt water and bleaching sun. In every painful exchange, they make thought of the other equal with thought for themselves and express themselves with civility and courtesy; these are the bedrock of their personalities:
Later the correspondent spoke into the bottom of the boat.
"Billie!" There was a slow and gradual disentanglement. "Billie, will you spell me?"
"Sure," said the oiler. [...]
it was but a moment before he heard a voice call his name in a tone that demonstrated the last stages of exhaustion. "Will you spell me?"
Shared Morality of Humanity: What they say and do when they finally approach safety, after the fears and dreads and shocks are realized or past, comes from their essential moral feelings since there is nothing left. We see the captain think of the cook's welfare and the correspondent's safety. We see the correspondent relinquish help to rescue the oiler. These are sincere, no veneer, no brute showing;
"I am going to drown? Can it be possible? ...."
[The man] gave a strong pull, and a long drag, .... The correspondent, schooled in the minor formulae, said: "Thanks, old man." But suddenly the man cried: "What's that?" He pointed a swift finger. The correspondent said: "Go."
Nothing in the text suggests a brute eclipsing a civilized veneer. Yet, if you must argue this idea, the episode with the canton flannel sea gull offers an opportunity. The gulls unnerve the men when the gulls come nearby, partly because they are scavengers of death. They may want to club one away when it threatens to land on the captain's head but, since motion is impossible, they instead shout to drive it off. This might possibly be said to indicate the hunter brute, breaking through a veneer of civilization who, faced with a weaker creature, wants to kill, not befriend, it.
At these times [the gulls] were uncanny and sinister in their unblinking scrutiny, and the men hooted angrily at them, telling them to be gone. ... The captain naturally wished to knock it away with the end of the heavy painter,....