The class of strong stochastic Runge-Kutta methods for stochastic differential equations (SDEs) due to Rößler is considered. Based on the order conditions of this class, we design a family of strong diagonally drift-implicit stochastic Runge-Kutta (DDISRK) methods of order one for solving Itô SDE systems with an m-dimensional standard Wiener process. We then explicitly provide the structure of the mean-square stability matrix of the new DDISRK methods for the general form of linear SDE systems. With some specific values of the coefficients for this family, particular DDISRK methods that have good stability properties are proposed. In order to check their convergence order and stability properties some numerical experiments are performed.