Different classical sources advance different arguments that relate to the serious question you raise. One can apply the values found in those sources to our present-day situation and, with some common sense, arrive at a reasonably balanced strategy.
The gemara[2] strongly criticizes one who gives all of the donations that he is required to present to kohanim to just a single kohen. The Shulchan Aruch[3] extends this rule to tzedaka, saying that one should not give it all to only one poor person. Several other sources follow this direction, but there seem to be two different reasons provided for dispersing tzedaka among several recipients. The Bach,[4] in explaining why we must give matanot la’evyonim to at least two people, says that it is important to bring sustenance to as many people as possible. The Rambam, commenting on the mishna in Pirkei Avot[5] that says that “it all follows the amount of action,” says that one’s attribute of generosity is developed better by performing more smaller acts of giving than by giving larger sums fewer times. Thus, while both sources encourage giving to multiple recipients, the former is concerned with the recipients’ physical welfare, whereas the latter focuses on the donor’s spiritual wellbeing.
On the other hand, there are sources that stress the importance of giving to fewer recipients in the hope of helping a person or organization reach the position of “dei machsoro” (enough to fill his needs).[6] That logic is stronger when the alternative is giving to so many people that no one gets a donation that significantly changes his predicament.[7] However, when one can help more people reach at least their most basic needs as opposed to helping fewer people attain less critical provisions, then the basic level of the many supersedes our desire to fulfill the preferred mitzva of dei machsoro. (The Chatam Sofer[8] says that one gives to his city’s poor before others only when they are in an equivalent situation to those of other cities; one does not make one set of paupers a few levels better off than another.) There is another modern consideration that makes it preferable to give larger sums to fewer institutions. Each donation costs money for a recipient organization (and the donor’s tzedaka account, if he has one), including banking fees, mailings, and secretarial work. If one mails an organization a $5 check, he has likely brought no net gain. (In contrast, giving a $5 cash donation to a door-to-door collector is an outright net gain for someone who already paid the expenses that brought him to the door).
We should also consider that most significant donations are given to organizations that assist many individuals. This has a few advantages. One check reaches a large number of recipients, fulfilling the Bach’s concern. That a donation may not change any individual’s situation is not so pertinent, as the nature of organizational tzedaka campaigns is to obtain many contributions so that, at the end, they can help many people significantly. On the other hand, to give all of one’s money to one source, even one that distributes to many needy people, seems to be undesirable from the Rambam’s perspective. What does it do to one’s neshama when he receives fifty requests from worthy causes, representing different populations and needs, and throws forty-nine into the garbage?[9] It is also possible that the organization with the most attractive campaign gets a disproportionate amount, and one who gives only to it misses out on entire groups in dire straits. It is thus healthy to “connect” with several causes and institutions, while trying to ensure that the amounts given are large enough to be helpful. As usual, balance is paramount.