I think this is a really important question, but I am not sure there is solid halachic guidance on this question. Let me share some of what I know and suggest where I think it points.
There were traditions, in the times of the Talmud and in medieval times, that certain decisions were made half by money (meaning: the rich people of the town or synagogue got half the votes on how to proceed) and half by population (so each townsperson got a vote). That seems to me not an unreasonable baseline assumption: to recognize that those supporting a town or synagogue do, in fact, deserve more of a say than those who merely populate it. I remember noticing, years ago, how so many Jewish communities rely on 5-10 top donors for a disproportionate percentage of their financial support; I remember, personally, thinking it was unsustainable, but being assured that was the way the world worked.
When it comes to a community, that system makes some sense, since the donors have two claims to their greater influence: first, by virtue of their wealth, we can at least suspect they know how to operate successfully in their local environments and, two, since they live there, they have to bear-- with the rest of the community-- the results or consequences of their decisions.
Once we turn to institutions, that gets more complicated, because donors' claim to expertise is much more tenuous. I don't imagine, for example, that when people give hundreds of millions of dollars to universities or medical centers, that they also think they know exactly how to actualize their hopes for the institution (they don't draw up the architectural plans themselves). At the same time, they rarely just walk up to an institution and say, here's the money, do whatever you want with it.
I think, most productively, it's a negotiation (not in the tense sense of each side trying to squeeze the other, but in the sense of people of goodwill, with differing perspectives, trying to work together to get to the best result). The donor, with the money, has a sense of what matters enough to him/her to support, and the institution-- hopefully staffed by experts, who can show options for where the money can be put to best use-- tries to find a niche that the donor will feel productive supporting. After the broad outlines, though, the institution's job is to put the money to good enough use, probably with repeat consultations with the donor, that everyone walks away happy with what they accomplished.
That is true, I think, without any Jewish aspect to it. Since this is a Jewish values website, though, I would add two more pieces: 1) Humility, a vital Jewish value, even for those of us who are told by the world that they are masters of the universe. The fact that I have 10 or 50 million dollars to donate to a cause (or a billion) cannot be allowed to instill arrogance. The attitude of the donor, ideally, should be that God has entrusted him/her with this money to put to best possible use in the general cause of making this the kind of world God "hoped" for when God created it. Properly experienced, humility means that the donor knows s/he is the steward of this money, and bears a sacred responsibility to find its best use. That means the donor neither abdicates responsibility for what happens to it (Institution X seems good; take the money and do whatever you want with it) nor loses sight of his or her own limitations of knowledge and understanding. That half of the puzzle should mean, I think, that the donor turns to 2) Experts-- the professional leadership of charitable institutions, often, are skilled professionals who know what is doable and how best to do it. When those people are also Jewishly knowledgeable, they often are also able to articulate how their mission advances God's presence in the world.
Putting all that together, I think donors should see their money as a tool they have been granted to make the world a better place. Sometimes, they might trust the leadership of an organization enough to just give the money freely, sometimes they'll want to be involved enough to insure that the money is put to uses of which they can feel proud. And, the most Jewish part of the whole discussion, to me, is that all the parties-- the donors and the professionals-- remember that each of them needs to fulfill their tasks with appropriate humility, and with the awareness of which areas are those of their expertise, and which are those where they would be well-advised to seek other opinions.
Answered by: Rabbi Gidon Rothstein