Biomolecular condensates participate in diverse cellular processes, ranging from gene regulation to stress survival. Bottom-up engineering of synthetic condensates advances our understanding of the organizing principle of condensates. It also enables the synthesis of artificial systems with novel functions. However, building synthetic condensates with a predictable organization and function remains challenging. Here, we use DNA as a building block to create synthetic condensates that are assembled through phase separation. The programmability of intermolecular interactions between DNA molecules enables the control over various condensate properties including assembly, composition, and function. Similar to the way intracellular condensates are organized, DNA clients are selectively partitioned into cognate condensates. We demonstrate that the synthetic condensates can accelerate DNA strand displacement reactions and logic gate operation by concentrating specific reaction components. We envision that the DNA-based condensates could help the realization of the high-order functions required to build more life-like artificial systems.