Simon Kuper, whom I like a lot, says: "Marx. like Freud, has finally died." This is right in one sense. Big teleological theories of history have been discredited; the only people who believe them now are silly centrists and bosses who believe the future is foreseeable and who prate about "modernization" as if it were an uncontestable idea.
However, as you'd expect, I think Marx is still very relevant today. Take four examples:
1. Why have governments encouraged financialization rather than financial democracy? Why were they keener to bail out banks than steel firms? Why did central banks conduct QE with some inegalitarian effects rather that a more egalitarian (and effective) helicopter drop? Why has austerity borne harder upon workers than the rich? One possibility is that the state is not a benevolent agent seeking to maximize a social welfare function but is instead, as Marx said, "a committee for managing the common affairs of the whole bourgeoisie."
2. Why are we in secular stagnation? One possibility is that, as Marx said, the rate of profit has declined (pdf), thus weakening the motive to invest. Another (not exclusive) possibility is that the fear of future technical progress deters investment, by increasing the fear that current investments will be on the wrong side of creative destruction.
3. Both these possibilities suggest that capitalism has become an obstacle to technical progress - that, as Marx said:
At a certain stage of development, the material productive forces of society come into conflict with the existing relations of production.
This might be true in another way: the inequality that capitalism fosters retards productivity growth.
4. Why has the relative position of low-paid workers worsened since the 70s? A big reason is that they have suffered from technical change and globalization. As Marx recognized, socio-technical change doesn't just increase the potential for human well-being; it also has class-biased effects.
In these senses, Marxian questions are relevant today: how does social change or government policy affect classes differently? Are individual choices consistent with aggregate well-being or not? Do capitalistic property rights increases economic growth or hold it back?
I'd add something else. Marx asked, and answered, a fundamental question: what is the point of economic life? For him, it was to increase real freedom and self-realization. Capitalism, he said, doesn't do this but instead alienates us:
What, then, constitutes the alienation of labor?
First, the fact that labor is external to the worker, i.e., it does not belong to his intrinsic nature; that in his work, therefore, he does not affirm himself but denies himself, does not feel content but unhappy, does not develop freely his physical and mental energy but mortifies his body and ruins his mind. The worker therefore only feels himself outside his work, and in his work feels outside himself. He feels at home when he is not working, and when he is working he does not feel at home. His labor is therefore not voluntary, but coerced; it is forced labor. It is therefore not the satisfaction of a need; it is merely a means to satisfy needs external to it...The external character of labor for the worker appears in the fact that it is not his own, but someone else’s, that it does not belong to him, that in it he belongs, not to himself, but to another.
As Jon Elster says: "Marx himself condemned capitalism mainly because it frustrated human development and self-actualization."
Again, this is still relevant today. As Bryson and Mackerron have shown, people feel unhappy whilst they are working. Even those of us in comfortable well-paid jobs look forward to retirement, setting up on our own, or escaping from the rat race.
All this raises the question: why, then, are so many people loath to acknowledge Marx's relevance?
One answer is that Marx has been discredited by his followers - not just the barbarism of Soviet dictatorship but also the dogmatism and sectarianism of some of today's Marxists.
Another answer is that there is a tendency - popularized perhaps by Kolakowski's influential Main Currents of Marxism - to think of Marx as an obscurantist Hegelian. To some extent, he was: we are all victims of our education. But there is another Marx. If you start from Engels' Condition of the Working Class in England and start reading Capital not from the beginning but from chapter 10, another Marx emerges - one whose thinking was rooted in empirical facts about the working lives of the worst off and in an urge to improve these. It is this Marx which is still relevant today.