Implementing application-specific computation and control tasks within a biochemical system has been an important pursuit in synthetic biology. Most synthetic designs to date have focused on realizing systems of fixed functions using specifically engineered components, thus lacking flexibility to adapt to uncertain and dynamically-changing environments. To remedy this limitation, an analog and modularized approach to realize reconfigurable neuromorphic computation with biochemical reactions is presented. We propose a biochemical neural network consisting of neuronal modules and interconnects that are both reconfigurable through external or internal control over the concentrations of certain molecular species. Case studies on classification and machine learning applications using the DNA strain displacement technology demonstrate the effectiveness of our design in both reconfiguration and autonomous adaptation.