Paul Dacre's charge of the BBC's being Marxist must surely be intended as a joke. Though I don't believe there's any single, authoritative, defining characteristic of Marxism, there are at least ideas that have been important within the tradition. You'd expect any Marxist to subscribe to a few of these, if not necessarily all of them: that an understanding of history and society depends crucially on focusing on the 'economic' dimension of social life; an emphasis on class as a key determinant in politics; the view that through most of human history the production of wealth has been based on the economic exploitation of the labour of the 'direct producers', and that this is unjust; the 'labour theory of value'; a belief that capitalism is permanently unstable and that it creates the preconditions for a new and better form of - classless - society; the idea that the modern working class will be the primary agent of bringing such a society about; more or less egalitarian ethical precepts; and so forth.
In explaining his charge Paul Dacre says nothing about any of this. True, what he alleges is that the Beeb 'exercises a kind of "cultural Marxism"' [My emphasis]. But all he seems to mean by this is that there's a leftist tilt to the outlook of most of those working for the corporation. Even when Marxism was much more influential as a set of ideas than it is now, it was never co-extensive with the left. And there's nothing in Dacre's claims that adds up to a credible definition of 'cultural Marxism'.