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We introduce a flexible framework based on response theory and score matching to suppress spurious, noncausal dependencies in reduced-order neural emulators of turbulent systems, focusing on climate dynamics as a proof-of-concept. We showcase the approach using the stochastic Charney-DeVore model as a relevant prototype for low-frequency atmospheric variability. We show that the resulting causal constraints enhance neural emulators' ability to respond to both weak and strong external forcings, despite being trained exclusively on unforced data. The approach is broadly applicable to modeling complex turbulent dynamical systems in reduced spaces and can be readily integrated into general neural network architectures.