Are people basically selfish?
On a normal understanding of selfishness, it doesn't seem to me that 'people are basically selfish' is a good reason for scepticism about the point of making moral claims. Here's why.
Suppose that you live with someone who selfishly never does any cleaning. That doesn't stop you saying that your housemate ought to clean more often. If the housemate were to reply by saying "look, I'm just fundamentally selfish!", that wouldn't wash at all. You wouldn't say: "Oh, okay, fine—I'll stop saying that you ought to clean more often."
If that's right, why should the fact—if it is a fact—that people are fundamentally selfish give us any reason to stop making moral claims?
But in any case, it doesn't look like it is a fact. Many people do things for others with no expectation of a net gain for themselves. For example, they hand in wallets when they could get away with keeping them, they help people who have hurt themselves, they give directions to strangers, they give money to charity, and so on. People sometimes even sacrifice their lives for the sake of others.
Sometimes, in reply to this, it's argued that even those who sacrifice their lives get a warm glow of satisfaction as they throw themselves in front of the bullets, or something like that. (And the same warm glow prompts the return of the wallet, the assistance to the needy, etc.) So, they turn out to be doing it for selfish reasons after all.
Now, that's certainly not how things seem. When you help someone who's hurt, for example, it typically seems to you as if you're doing it because the person needs help—because you'll do good by helping. You may indeed get a nice warm glow afterwards, but it's usually a mistake to characterise that warm glow as your conscious motivation for helping in the first place.
That being so, the argument has to be that we're actually deceiving ourselves, and that the warm glow is really our motivation, even though we don't think that it is. But where's the evidence for this? I'm not aware of any. And if you don't have any yourself, then the warm-glow claim looks as if it's the product of your theory that everyone is selfish, rather than an argument for that theory. In that case, your theory remains unsupported by argument.
(A final question for those who are still sceptical: if it's true that we get a nice warm glow as a result of acting benevolently, and that's why we help others, doesn't the fact that it's helping others that gives us the glow rather suggest that we're constitutionally unselfish?)
Selfishness and evolution
In fact, what I think people who make the 'selfishness' argument are really getting at is a kind of evolutionary argument. The idea, roughly, is that our drive to survive is fundamental, and that everything we do must therefore be ultimately directed at survival.
But we need to be careful about interpreting this supposed drive for survival as a form of selfishness. Selfishness, in the ordinary sense of the word, may well be a bad survival strategy. After all, we've evolved to be moral creatures who put pressure on each other not to be selfish. Being selfish doesn't make you attractive either to potential mates or to potential co-operators. So, in one way, we've presumably evolved to be unselfish.
In that case, it's not true that humans are fundamentally selfish after all—at least, not in any sense that's inconsistent with their being morally motivated creatures.
In conclusion: in the ordinary sense of selfish, it's not at all clear that people are fundamentally selfish. And in the special evolutionary sense of 'selfish' that means 'driven to survive', it's not clear that being selfish is inconsistent with being moved by moral thinking. So, it seems that making moral claims isn't pointless for this kind of reason after all. If people believe them, they'll have an effect.