Making the studied subject useful at something would be a good start.
In the specific case, the definition of "pure mathematics" deserves an historical introduction, to make clear it has nothing to do with the problem-solving mathematics. It is a subject that was founded only in the thirties by the Bourbaki group in France. The group was chasing the goal of finding a complete and coherent foundation of mathematical analysis, since few members of the groups had found some counterexamples in the Gourstat's book used back then.
The complete detachment from physics, and consequent creation of theories descending only from axioms, where no single problem is ever solved, is what characterises the "pure" approach established back then.
Fortunately or not, Goedel incompleteness theorem wiped out the possibility of the coherent and complete theory sought by Bourbaki for good. Despite this event, the branch of "pure mathematics", as it fits very well with the academic environment and the need of a hierarchy not based on merit (if no problem is solved, who will tell who deserves a promotion and how does not), is still a subject per se, and it had become as well as a pedagogical methodology (and a detrimental one, according to many).
Interesting to notice that there have never been such a thing as "pure mathematics" before Bourbaki, having maths always something applied to solve real problems for the entire history of humanity: finding algorithms, and generalisations only when it was possible to apply a solution found in one instance to solve a wider range of problems. Archimedes, Gauss or Euler would have not understood the question: "are you a pure mathematician?".
To the commenters distressed at the idea of a university without a math department, they can be reassure that "pure mathematics" has nothing to do with the mathematics they have studied for the A-level, or the mathematics they know of as a tool to solve problems.
Besides the theoretical groundlesness of pure mathematics, as mathematically proven by Goedel, the "pure" mathematics studies certainly have a detrimental effects on their students. I recently interviewed a candidate with a pure mathematics background, graduated with full marks with a thesis on a subject he was not able to explain during the interview, who turned out to have never heard of graph theory. Also he heard of the existence of differential equations, but as something for another department, not for him, so he had never solved one.
There is the curious situation of an increasing amount of industries thirsty for mathematical talents, as problem solvers and algorithms developer, and the "pure" mathematics department is producing graduates with the title of mathematicians who are crippled by too much useless theory, lost in a machinery of details, and unable to solve anything or compute anything.
Leaving the abstract manipulation of symbols, and teaching Algorithms and Differential Equations ("with their applications", as purists like to put it) instead of axiomatic theories closed in boxes, would be the best strategy to keep the maths departments alive, and provide students with capabilities other than knowledge.
I must have touched a point, since all my posts have been downvoted shortly after writing this one. Bringing down the reputation of the author instead of talking about ideas, now, that's a strategy never used before! .-)