Climate change is altering the epidemiological landscape of vector-borne diseases, with dengue fever and malaria presenting particular challenges for public health systems in tropical and subtropical regions. This study develops and analyses a climate-sensitive compartmental model based on a system of ordinary differential equations (ODEs) that incorporates temperature, rainfall, and humidity as dynamic inputs governing mosquito population biology and disease transmission. The basic reproduction number (R₀) is derived analytically, and local stability conditions for both the disease-free equilibrium (DFE) and the endemic equilibrium (EE) are established. Numerical simulations conducted over a three-year horizon under baseline and perturbed climate scenarios reveal that temperature is the dominant driver of outbreak intensity (PRCC = 0.75), with a +2°C anomaly sufficient to push R₀ above unity and elevate outbreak probability beyond 80% in high-density urban settings. Dengue transmission peaks at 37.5°C with 120–140 mm monthly rainfall, while malaria transmission is optimal at 25°C with relative humidity exceeding 70%. These findings quantify critical climatic thresholds and underscore the need to integrate climate projections into epidemiological early-warning systems and adaptive public health policy.