A shipowner was about to send to sea an emigrant-ship. He knew that she was old, and not overwell built at the first; that she had seen many seas and climes, and often had needed repairs. Doubts had been suggested to him that possibly she was not seaworthy. These doubts preyed upon his mind, and made him unhappy; he thought that perhaps he ought to have her thoroughly overhauled and refitted, even though this should put him at great expense. Before the ship sailed, however, he succeeded in overcoming these melancholy reflections. He said to himself that she had gone safely through so many voyages and weathered so many storms that it was idle to suppose she would not come safely home from this trip also. He would put his trust in Providence, which could hardly fail to protect all these unhappy families that were leaving their fatherland to seek for better times elsewhere. He would dismiss from his mind all ungenerous suspicions about the honesty of builders and contractors. In such ways he acquired a sincere and comfortable conviction that his vessel was thoroughly safe and seaworthy; he watched her departure with a light heart, and benevolent wishes for the success of the exiles in their strange new home that was to be; and he got his insurance-money when she went down in mid-ocean and told no tales.
Clifford from this draws the conclusion that we would recognize the ship captain as responsible for the deaths of those who died and that, in particular, we would say that "the sincerity of his conviction can in no wise help him, because he had no right to believe on such evidence as was before him."
The example is often quoted or alluded to in discussions of the ethics of belief, due to Clifford; but always uncritically, I think. More careful reasoners should pause here, because Clifford's ethical analysis is just obviously bad. First, like everything in the essay, the condemnation is stated in an exaggerated manner; on most ethical theories we could not say "that he was verily guilty of the death of those men" without qualification. Negligent omission, even egregiously culpable negligent omission, does not work like deliberate commission, and does not interact with guilt in the same way. Further, Clifford is playing a rhetorical game in his parting shot at the sea captain; told that someone "got his insurance-money...and told no tales", we would usually take this to suggest that the matter was in fact more deliberate than the sea captain's actions are actually presented as being -- that, in fact, he was at least half-angling toward the insurance money to begin with, particularly given the prior emphasis on expense. But what Clifford needs for his argument is really a clean case -- someone believing badly without the additional unsavory suggestion of things like greed. He needs a sea captain who is, as he previously said, genuinely benevolent, and whose only flaw is this. Otherwise you get cross-interference that blunts the usefulness of the example for the purpose of showing that there are obligations of belief in particular, and not just obligations not to be greedy.
Worse, the ship captain's belief is simply irrelevant here. The sea captain's sincerity of belief does not help him, to be sure, but it is not because "he had no right to believe", but because the belief in question doesn't matter. We condemn the sea captain not because he believed badly; we condemn the sea captain because, given his doubts, he had responsibilities regardless. Likewise, this is why it doesn't matter whether his belief turns out to be right or not: not because he had no right to belief, but because his responsibilities didn't depend on that belief at all. His responsibilities are based on the warning-flags that had been raised; it didn't matter what he in fact believed about the ship.
What is more, if we look at the timeline here, we find that it is poorly suited for Clifford's ultimate point. The timeline is as follows:
(1) The shipowner is preparing to send the ship, knowing that it is old, flawed, and often in need of repair.
(2) He has doubts that the ship is not seaworthy.
(3) He thinks that perhaps he has an obligation to have her thoroughly overhauled and refitted.
(4) He tries to talk himself out of this and tries to dismiss some of his worries.
(5) He comes to have a sincere and comfortable conviction that the vessel is seaworthy.
(6) He sends the ship off with light heart and benevolent wishes.
The first thing to note is that the sea captain starts out believing that he might have an obligation, and then actively tries to talk himself out of that until he succeeds at (5). This is important because his actual obligation as a sea captain begins at (2), at which point he still suspects there might be a problem. All that the rest of the case shows, as far as the ethics of it, is that (3)-(6) don't affect this obligation at all -- he still has the same obligation throughout. Further, the real problem with the sea captain's final belief is not that he fails to believe in accordance with the evidence; it is that, thinking he had an obligation, one he actually had, he deliberately tried to convince himself that he didn't. This is where the appearance of the case being one of 'ethics of belief' comes from; it is the unethical nature of the motivation on which he is trying to convince himself not to believe what he does. But this is not about the belief; this is about trying to give yourself a belief with a motivation that is already and independently unethical.
And if we needed another reason to be skeptical of this commonly repeated case, the case is poorly suited just in itself for showing that Clifford's principle (it is wrong to believe on insufficient evidence) is true, because very little evidence is actually mentioned -- the ship is old, not all that well built, and has needed repairs before, and, on the other side, that she has gone safely through a lot of voyages and weathered a lot of storms. Everything else is just referred to as doubt and suspicion. Evidence doesn't play much of a role at all in our assessment of the moral situation -- indeed, once the sea captain has significant doubts, he already has at least some responsibility to double-check and take precautionary steps, even if he's nervously overreading the evidence and a more reasonable assessment of the ship would judge it to be just fine.
Thus the belief, as such, is irrelevant to the moral judgment; the evidence is not given much of a role in the scenario; and the ethical features of the scenario are not strongly tied to either. It's just a poor case.