Upper atmospheres of Hot Jupiters are subject to extreme radiation conditions that can result in rapid atmospheric escape. The composition and structure of the upper atmospheres of these planets are affected by the high-energy spectrum of the host star. This emission depends on stellar type and age, which are thus important factors in understanding the behaviour of exoplanetary atmospheres. In this study, we focus on Extrasolar Giant Planets (EPGs) orbiting K and M dwarf stars. XUV spectra for three different stars – ∊ Eridani, AD Leonis and AU Microscopii – are constructed using a coronal model. Neutral density and temperature profiles in the upper atmosphere of hypothetical EGPs orbiting these stars are then obtained from a fluid model, incorporating atmospheric chemistry and taking atmospheric escape into account. We find that a simple scaling based solely on the host star’s X-ray emission gives large errors in mass loss rates from planetary atmospheres and so we have derived a new method to scale the EUV regions of the solar spectrum based upon stellar X-ray emission. This new method produces an outcome in terms of the planet’s neutral upper atmosphere very similar to that obtained using a detailed coronal model of the host star. Our results indicate that in planets subjected to radiation from active stars, the transition from Jeans escape to a regime of hydrodynamic escape at the top of the atmosphere occurs at larger orbital distances than for planets around low activity stars (such as the Sun).