The proliferation of mobile phones in low- and middle-income countries has suddenly and dramatically increased the extent to which the world's poorest and most vulnerable populations can be observed and tracked by governments and corporations. Millions of historically "off the grid" individuals are now passively generating digital data; these data, in turn, are being used to make life-altering decisions about those individuals -- including whether or not they receive government benefits, and whether they qualify for a consumer loan. This paper develops an approach to implementing algorithmic decisions based on personal data, while also providing formal privacy guarantees to data subjects. The approach adapts differential privacy to applications that require decisions about individuals, and gives decision makers granular control over the level of privacy guaranteed to data subjects. We show that stronger privacy guarantees typically come at some cost, and use data from two real-world applications -- an anti-poverty program in Togo and a consumer lending platform in Nigeria -- to illustrate those costs. Our empirical results quantify the tradeoff between privacy and predictive accuracy, and characterize how different privacy guarantees impact overall program effectiveness. More broadly, our results demonstrate a way for humanitarian programs to responsibly use personal data, and better equip program designers to make informed decisions about data privacy.