We introduce new approaches aimed at improving the stability properties of traditional explicit numerical methods used to solve systems of stochastic differential equations (SDEs), possibly arising from the semi-discretization in space of stochastic partial differential equations (SPDEs). The key innovation lies in modifying the standard coefficients of explicit methods by incorporating matrices that depend on the Jacobian of the drift term in the SDE. The proposed methods are constructed by enhancing established explicit schemes with TASE (Time-Accurate and Highly-Stable Explicit) operators, in order to achieve better stability. Nevertheless, as the methods require the inversion of Jacobiandependent matrices at each time step, the resulting schemes are linearly implicit methods. We derive TASE stochastic Runge-Kutta methods with strong convergence orders of p = 1 and p = 1.5 and we carry out numerical experiments to confirm the theoretical results. This research activity falls within the activities of PRIN-MUR 2022 project 20229P2HEA ”Stochastic numerical modelling for sustainable innovation”, CUP: E53D23017940001, granted by the Italian Ministry of University and Research within the framework of the Call relating to the scrolling of the final rankings of the PRIN 2022 call.