You probably understand that a full answer to this question would require an enormous amount of space (and time), and that it would involve all kinds of diversions into controversial topics. But since you're curious, I will try to provide a cursory outline of my views that are relevant in this context.
About a century ago -- and perhaps even earlier -- one could notice two trends in the public perception of science, caused by its immense practical success in providing all sorts of world-changing technological marvels. First, this success had given great prestige to scientists; second, it had opened hopes that in the future science should be able provide us with foolproof guidance in many areas of human concern that had theretofore been outside the realm of scientific investigation. The trouble with these trends was that around this time, dreams and hopes fueled by them started to seriously drift away from reality, and as might be expected, a host of pseudo-scientific bullshit-artists, as well as political and bureaucratic players with ready use for their services, quickly arose to exploit the opportunities opened by this situation.
This has led to a gradually worsening situation that I described in an earlier LW comment:
The trouble nowadays is not that governments are not listening to scientists (in the sense of people officially and publicly recognized as such), but that the increased prominence of science in public affairs has subjected the very notion of "science" to a severe case of Goodhart's law. In other words, the fact that if something officially passes for "science," governments listen to it and are willing to pay for it has led to an awful debasement of the very concept of science in modern times.
Once governments started listening to scientists, it was only a matter of time before talented charlatans and bullshit-artists would figure out that they can sell their ideas to governments by presenting them in the form of plausible-looking pseudoscience. It seems to me that many areas have been completely overtaken by this sort of thing, and the fact that their output is being labeled as "scientific" and used to drive government policy is a major problem that poses frightful threats for the future.
This, in my view, is one of the worst problems with the entire modern system of government, and by far the greatest source of dangerous falsity and nonsense in today's world. I find it tragicomic when I see people worrying about supposedly dangerous anti-scientific trends like creationism or postmodernism, without realizing that these are entirely marginal phenomena compared to the corruption that happens within even the most prestigious academic institutions due to the fatal entanglement of science with ideology and power politics, to which they are completely oblivious, and in which they might even be blindly taking part. Just the thought of the disasters that our governments might wreak on us by pushing policies guided by this pseudo-scientific input should be enough to make one shiver -- especially when we consider that these processes typically operate on bureaucratic auto-pilot, completely outside of the scope of politics that gets public attention.
Whether or not you agree with this, I hope it clarifies the reasons why I have such strong interest in topics of this sort.
In some ways, things have gotten better, not worse. Both communism and Nazism claimed scientific backing. I don't see anything like that on the horizon.
On the other hand, people became disenchanted with them because of disastrous results-- I don't think there's any public recognition of the poor quality of science they used.
tl;dr: Relativism bottoms-out in realism by objectifying relations between subjective notions. This should be communicated using concrete examples that show its practical importance. It implies in particular that morality should think about science, and science should think about morality.
Sam Harris attacks moral uber-relativism when he asserts that "Science can answer moral questions". Countering the counterargument that morality is too imprecise to be treated by science, he makes an excellent comparison: "healthy" is not a precisely defined concept, but no one is crazy enough to utter that medicine cannot answer questions of health.
What needs adding to his presentation (which is worth seeing, though I don't entirely agree with it) is what I consider the strongest concise argument in favor of science's moral relevance: that morality is relative simply means that the task of science is to examine absolute relations between morals. For example, suppose you uphold the following two moral claims:
First of all, note that questions of causality are significantly more accessible to science than people before 2000 thought was possible. Now suppose a cleverly designed, non-invasive causal analysis found that physically punishing children, frequently or infrequently, causes them to be more likely to commit criminal violence as adults. Would you find this discovery irrelevant to your adherence to these morals? Absolutely not. You would reflect and realize that you needed to prioritize them in some way. Most would prioritize the second one, but in any case, science will have made a valid impact.
So although either of the two morals is purely subjective on its own, how these morals interrelate is a question of objective fact. Though perhaps obvious, this idea has some seriously persuasive consequences and is not be taken lightly. Why?
First of all, you might change your morals in response to them not relating to each other in the way you expected. Ideas parse differently when they relate differently. "Teachers should be allowed to physically punish their students" might never feel the same to you after you find out it causes adult violence. Even if it originally felt like a terminal (fundamental) value, your prioritization of (2) might make (1) slowly fade out of your mind over time. In hindsight, you might just see it as an old, misinformed instrumental value that was never in fact terminal.
Second, as we increase the number of morals under consideration, the number of relations for science to consider grows rapidly, as (n2-n)/2: we have many more moral relations than morals themselves. Suddenly the old disjointed list of untouchable maxims called "morals" fades into the background, and we see a throbbing circulatory system of moral relations, objective questions and answers without which no person can competently reflect on her own morality. A highly prevalent moral like "human suffering is undesirable" looks like a major organ: important on its own to a lot of people, and lots of connections in and out for science to examine.
Treating relativistic vertigo
To my best recollection, I have never heard the phrase "it's all relative" used to an effect that didn't involve stopping people from thinking. When the topic of conversation — morality, belief, success, rationality, or what have you — is suddenly revealed or claimed to depend on a context, people find it disorienting, often to the point of feeling the entire discourse has been and will continue to be "meaningless" or "arbitrary". Once this happens, it can be very difficult to persuade them to keep thinking, let alone thinking productively…
To rebuke this sort of conceptual nihilism, it's natural to respond with analogies to other relative concepts that are clearly useful to think about:
"Position, momentum, and energy are only relatively defined as numbers, but we don't abandon scientific study of those, do we?"
While an important observation, this inevitably evokes the "But that's different" analogy-immune response. The real cure is in understanding explicitly what to do with relative notions:
To use one of these lines of argument effectively — and it can be very effective — one should follow up immediately with a specific example in the case you're talking about. Don't let the conversation drift in abstraction. If you're talking about morality, there is no shortage of objective moral relations that science can handle, so you can pick one at random to show how easy and common it is:
"Teen pregnancy / the spread of STDs is undesirable."
Question: Does promoting the use of condoms increase or decrease teen pregnancy rates / the spread of STDs?
"Married couples should do their best not to cheat on each other."
Question: Does masturbation increase or decrease adulterous impulses over time?
"Children should not be raised in psychologically damaging environments."
Question: What are the psychological effects of being raised by gay parents?
I'm not advocating here any of these particular moral claims, nor any particular resolution between them, but simply that the answer to the given question — and many other relevant ones — puts you in a much better position to reflect on these issues. Your opinion after you know the answer is more valuable than before.
"But of course science can answer some moral questions... the point is that it can't answer all of them. It can't tell us ultimately what is good or evil."
No. That is not the point. The point is whether you want teachers to beat their students. Do you? Well, science can help you decide. And more importantly, once you do, it should help you in leading others to the same conclusion.
A lesson from history: What happens when you examine objective relations between subjective beliefs? You get probability theory… Bayesian updating… we know this story; it started around 200 years ago, and it ends well.
Now it's morality's turn.