You can ask to see the financial records for any organization, just say that you want to assure yourself that they are in good shape. Most (though not all) recognized 501c3 organizations (those with budgets over a very minimal one) MUST share their IRS 990 with you on request. You can also find this information athttp://www.GuideStar.org
a site which provides information on most charities and foundations. And, in most states in the US, you can also call your State Department of State (sorry for the redundancy) and find out if a charity is registered with them.
Even if you check out all of this, I would urge that the only real way of checking an organization out is to start going to their events and meetings; start volunteering. By the way, if you are someone who has even a LITTLE financial management savvy, PLEASE consider volunteering to help them in that area!! Most, if not all, animal welfare organizations are sorely in need of that kind of help -- most definitely would LIKE to be good efficient operations that make the most of every donation, but often they do not have a volunteer willing to dig into the accounting work that takes!