There was once an island in which some of the inhabitants professed a religion teaching neither the doctrine of original sin nor that of eternal punishment. A suspicion got abroad that the professors of this religion had made use of unfair means to get their doctrines taught to children. They were accused of wresting the laws of their country in such a way as to remove children from the care of their natural and legal guardians; and even of stealing them away and keeping them concealed from their friends and relations. A certain number of men formed themselves into a society for the purpose of agitating the public about this matter. They published grave accusations against individual citizens of the highest position and character, and did all in their power to injure these citizens in their exercise of their professions. So great was the noise they made, that a Commission was appointed to investigate the facts; but after the Commission had carefully inquired into all the evidence that could be got, it appeared that the accused were innocent. Not only had they been accused of insufficient evidence, but the evidence of their innocence was such as the agitators might easily have obtained, if they had attempted a fair inquiry. After these disclosures the inhabitants of that country looked upon the members of the agitating society, not only as persons whose judgment was to be distrusted, but also as no longer to be counted honourable men. For although they had sincerely and conscientiously believed in the charges they had made, yet they had no right to believe on such evidence as was before them. Their sincere convictions, instead of being honestly earned by patient inquiring, were stolen by listening to the voice of prejudice and passion.
The introduction of the Commission means that this case does not suffer from the problem the sea captain case had with its failure to make actual evidence central to the scenario; while we do not know the evidence (on either side), we have evidence of evidence, and the question of evidence ends up being central to the case. So there's certainly an improvement. The mention of original sin and eternal punishment probably indicates that Clifford is thinking of the actual treatment of the so-called Rational Dissenters, or similar groups, in England. If we look at the timeline, we get:
(1) A suspicion gets abroad that these people are (legally or illegally) taking children from their families in order to indoctrinate them.
(2) Some islanders form a society to agitate about the matter.
(3) The society actively accuses and tries to injure the livelihoods and reputations of the people in question.
(4) A commission is appointed to inquire into the matter.
(5) The commission concludes: "Not only had they been accused of insufficient evidence, but the evidence of their innocence was such as the agitators might easily have obtained, if they had attempted a fair inquiry."
(6) The inhabitants conclude that the members of the society are not only to be distrusted but not to be counted as honorable.
The moral failure of the society in question arises in leaping to action ahead of inquiry, in (2)-(3), which is confirmed by the commission's judgment that if they had inquired, the evidence that they were wrong was easily available. Indeed, all of (4)-(6) is not more than confirmation of what can be determined by stage (3). This is why Clifford is surely right that it would not, ethically speaking, be much different if the Commission had happened to vindicate their conclusion but not their inquiry.
The obvious question (which Clifford recognizes) is why this situation should be characterized as "no right to believe" rather than "no right to act". Consider, for example, the following timeline of a different scenario:
(1) A suspicion gets abroad that the religious are taking children from their families in order to indoctrinate them.
(2) Some islanders, believing the suspicion, form a society to agitate for a commission to investigate the matter, and do nothing else but try to get a commission set up.
(3) A commission is appointed to inquire into the matter.
If the belief were really the linchpin of moral judgment here, one would expect these islanders, too, to be condemned as having "no right to believe". But this is certainly not what most people would say. Most people would not condemn these agitators at all. What really draws condemnation from the actual case is trying to destroy people socially; it's this that people would be most likely to say they had no right to do. One can ignore the common reaction, of course, although Clifford's own view of morality was that it gets its general authority from being an internalization of tribal agency (in the form of actions that help the tribe survive), so I don't think he himself could dismiss the question of what the actual tribal response would be. But we can perhaps give him a little wiggle-room here, since there are cases in which people do have a real "no right to believe" response -- for instance, if I believed you were a pathological liar on no real grounds, you would likely be indignant, and many responses to prejudice are at least sufficiently like "no right to believe" responses that we can grant that he is not seriously distorting things. But it's noticeable that you can massively affect the kind of moral judgment people would usually be expected to make without changing the belief, or how it was obtained, at all.
When Clifford addresses the question of why it should be "no right to believe" rather than "no right to act", however, he makes what I think is a fatal mistake. After conceding that it's right that, even given the belief, we have some power of choice, he says:
But this being premised as necessary, it becomes clear that it is not sufficient, and that our previous judgment is required to supplement it. For it is not possible so to sever the belief from the action it suggests as to condemn the one without condemning the other. No man holding a strong belief on one side of a question, or even wishing to hold a belief on one side, can investigate it with such fairness and completeness as if he were really in doubt and unbiased; so that the existence of a belief not founded on fair inquiry unfits a man for the performance of this necessary duty.
He then says a bit later:
And no one man's belief is in any case a private matter which concerns himself alone. Our lives our guided by that general conception of the course of things which has been created by society for social purposes. Our words, our phrases, our forms and processes and modes of thought, are common property, fashioned and perfected from age to age; an heirloom which every succeeding generation inherits as a precious deposit and a sacred trust to be handled on to the next one, not unchanged but enlarged and purified, with some clear marks of its proper handiwork. Into this, for good or ill, is woven every belief of every man who has speech of his fellows.
Let us concede for the moment that if you condemn acting in a certain way, you condemn believing in such a way that tends to lead you to act in that way. Most people would think this, like most of what we find in Clifford's essay, is exaggerated, but granting it doesn't really help Clifford at all. Likewise, most people would regard "no one man's belief is in any case a private matter" as an obvious exaggeration, as well. Clifford himself takes it quite seriously; since Clifford doesn't believe there are any self-regarding virtues, he can only get the strong moral conclusion he wants to draw about all beliefs by insisting that all beliefs are relevant to other-regarding virtues, and thus that every belief whatsoever is of moral significance for our social relations. But we can concede it for the moment, as well, because it also doesn't avoid the major problem.
The major problem is this. In order to avoid the objection that the ethical analysis should be "no right to act" rather than "no right to believe", Clifford has responded by saying that every belief has at least an indirect effect on action, and thus the condemnation of the action gives one the ground for condemning the belief. But the conclusion he wishes to draw on this basis is that it is always wrong to believe on insufficient evidence, and he cannot get it. Because the conclusion that follows is that beliefs are to be condemned not on how they were reached, but that they are to be condemned on the basis of the wrongness of the actions to which they tend. The principle this argument supports is that it is always wrong to believe things that tend toward doing morally bad things or having morally bad habits. This is not the same principle as Clifford's principle at all. This is a particularly serious problem given Clifford's own conception of morality as based on tribal survival, since there is no reason whatsoever to think that literally every kind of belief without evidence is dangerous for the tribe. I suspect this is why Clifford ends up emphasizing the importance of not getting into the habit of believing without evidence; although, of course, one can have develop more sophisticated habits than that. But even if we try to substitute some other morality, there is no serious candidate for ethics that takes believing only with sufficient evidence as so important that it overrides every other moral consideration; and given that, the question will always be, "Why don't some of these higher moral considerations sometimes allow or even require belief in advance of the evidence?" We get no answer on this ground from Clifford himself, despite his high-toned, exaggeratedly moralistic preaching. Ironically, he argues for his principle not on the basis of sufficient evidence but on the insufficiently supported ground that it is bad for society if we don't accept it. (This is, incidentally, part of why James argues against him the way he does in The Will to Believe; and it is the entire reason that Ward takes the approach he does against Clifford in The Reasonable Basis of Certitude.) It's notable, in any case, that in at least one case in the essay he slides between talking about believing on "unworthy reasons" and believing on "insufficient evidence", as if they were the same thing; his argument commits him to thinking they are, despite the fact that this is never established.
Thus Clifford's island agitators example is in some ways a better case than the sea captain scenario, but it also makes clear an overall problem with the way Clifford attempts to establish his principle, by raising even more forcefully and obvious the objection that our moral judgments seem concerned with the actions rather than the beliefs.