Lee Crawfurd asks that great question.
One way is by teaching, but my comments are focused on research by universities located in developing countries:
1) Universities located in developing countries might probably be in better position to influence policies in their own countries. Local knowledge is key. Following that logic probably one way in which research done in developed countries can have impact in developing countries is to include in the studies researchers from developing countries, they have access to local networks to disseminate and "translate" research findings.
2) It is a paradox, however, that many universities in developing countries underestimate the value of research. They see research, for example, as something that is very expensive, which of course can be, but it does not have to be. In fact, what matters is evidence to guide public policy, and some evidence may be better than nothing.
4) Before asking how universities can influence policy we should ask what kind of incentives should exist in developing countries for research to be done in the first place. In universities in the US there are competitive small-grants for faculty research, we are talking about grants of US$ 500 or 1000. There is very little of that sort, which stimulates academic competition, at universities in developing countries. That money can have large effects.
5) Recently I heard Luis Pazos [link in Spanish] a Mexican economist, who writes short books on policy issues, and he hands them to mexican public servants. It seems that by giving short research reports directly to policy makers a lot can be achieved. He said that congressmen for example want to read accessible literature to make decisions, but unfortunately they don't find easily useful things to read. Pazos has written around forty short books with the help of only one research assistant. Imagine what universities could do . . .
6) I think Colombia has now universities that are pushing policy-oriented research, such as Universidad de los Andes. We should look closely at those universities, which can become models for other schools in developing countries.
7) We should show that countries where universities do research actually have better social/economic performance (have better policies) in order to convince university leaders that research pays-off.
8) The media in developing countries should see the value in disseminating and commenting on academic research. There is very little of that in some countries.
9) Only few universities grant tenure in developing countries . . .