Anisotropy in the optical and electrical properties of many transition metal dichalcogenides provides a great variety of possibilities for device applications. The present work attempts to exploit the anisotropy in photoresponse of pure and Co-incorporated WSe2 crystals. Energy-dispersive analysis of X-rays affirmed the elemental composition of the crystals grown by the direct vapour transport technique. The surface morphology of the crystals was studied by optical microscopy and scanning electron microscopy, both of which confirmed the layered structure of the same. Layers of the samples and the lattice fringes were concluded under high-resolution transmission electron microscopy. The crystalline nature and hexagonal phase of the crystals were confirmed from X-ray diffraction, further highlighted in the results observed from the selected area electron diffraction pattern. Anisotropic charge conduction properties under illumination due to the layered structure of the sample were analysed. The anisotropy in the photoresponse was confirmed from the current-voltage (I-V) characteristics, which were further explored from its time-dependent photoresponse. The crystals proved to be self-powered photodetectors with higher values of photocurrent (μA), photoresponsivity (μAW−1) and photodetectivity (~107-108 Jones) recorded for contacts taken parallel to the crystallographic c-axis of the crystals. The effect of Co in improving the photoconductivity of WSe2 has been analysed based on the photoresponse recorded for pure and Co-incorporated WSe2. The present work also illustrates the effect of intensities of illumination and wavelengths of incident radiations in the photocurrent. Photocurrent in μA range observed in the absence of biasing voltage makes the crystals suitable candidates for self-powered photodetection applications.