Two excellent websites, Charity Navigator and GuideStar, can provide detailed information on how charities use donations. Charity Navigator rates organizations with stars from one to four. If the organization you're researching doesn't have four stars, don't give up on it. There are many reasons why it might not have the highest rating that don't have anything to do with how donations are handled. GuideStar offers copies of the 990s (the IRS tax return for nonprofit organizations) for hundreds of thousands of nonprofits. There are other organizations that provide information on charitable institutions; just Google "something like charity watchdog groups."
If you cannot find the organization you're researching on a watchdog site, contact the organization. Ask them to send you a copy of their 990 and any other pertinent data. If they refuse, find another similar organization that will.
Once you have the financials for an organization, look to see what percentage of donations goes to programs and what percentage goes to administrative expenses. According to CharityWatch (another watchdog organization), at least 60% of funds should go towards programs. A highly efficient organization will spend 75% or more on programs. In addition, look to see that it has low fund-raising and administrative costs. Although it does cost money to run a nonprofit, to pay staff and purchase office space, equipment, and supplies, and to purchase marketing materials, etc. for fundraising, these costs should not be a large percentage of monies spent. When I give, I want to make sure that the money I sent is actually going to help the individuals in need and not to give a big paycheck to the CEO.
I recently learned that organizations such as churches, synagogues, mosques, and other houses of worship are not limited by designated giving. That is, when someone gives money and specifically states that money is to go to missions or the benevolent fund, a faith organization can actually spend that money any way the pastor, rabbi, or administrative council or board deems fit. Therefore, if you do not automatically receive a budget for your house of worship, ask that one be provided so you can see how tithes and offerings are spent.
There are many other things you can do when making a decision to support a nonprofit organization. However, these tips should help point you in the right direction when your heartstrings feel a tug and you want to help.