This study is aimed at developing a fibre-reinforced polymer composite with a high bio-based content and to investigate its mechanical properties. A novel basalt fibre-reinforced polymer (BFRP) composite with bio-based matrix modified with different contents of star-like n-butyl methacrylate (n-BMA) block glycidyl methacrylate (GMA) copolymer has been developed. n-BMA blocks have flexible butyl units, while the epoxide group of GMA makes it miscible with the epoxy resin and is involved in the crosslinking network. The effect of the star-like polymer on the rheological behaviour of the epoxy was studied. The viscosity of the epoxy increased with increase in star-like polymer content. Tensile tests showed no noteworthy influence of star-like polymer on tensile properties. The addition of 0.5 wt.% star-like polymer increased the glass transition temperature by 8.2 °C. Mode-I interlaminar fracture toughness and low-velocity impact tests were performed on star-like polymer-modified BFRP laminates, where interfacial adhesion and impact energy capabilities were observed. Interlaminar fracture toughness improved by 45% and energy absorption capability increased threefold for BFRP laminates modified with 1 wt.% of star-like polymer when compared to unmodified BFRP laminates. This improvement could be attributed to the increase in ductility of the matrix on the addition of the star-like polymer, increasing resistance to impact and damage. Furthermore, scanning electron microscopy confirmed that with increase in star-like polymer content, the interfacial adhesion between the matrix and fibres improves.