We study stochastic proximal-point method applied to a convex-composite optimization problem, where the objective function is given as the sum of two convex functions, one of which is smooth while the other is not necessarily smooth but has a simple structure for evaluating the proximal operator. The main goal is to investigate a trade-off between the choice of a constant stepsize value and the speed at which the algorithm approaches the optimal points. We consider the case of a strongly convex objective function and make the most standard assumptions on the smooth component function and its stochastic gradient estimates. First of all, we analyze the basic properties of the stochastic proximal-point mapping associated with the procedure under consideration. Based on these properties, we formulate the main result, which provides the explicit condition on the constant stepsize for which the stochastic proximal-point method approaches a σ-neighborhood of the optimal point in expectation, where the parameter σ > 0 is related to the variance of the stochastic gradient estimates. Moreover, the rate at which the σ-neighborhood attracts the iterates is geometric, which allows us to estimate the number of iterations the procedure needs to enter this region (in expectation).