In the paper, we propose a construction of new cubic spline-wavelet bases on the unit cube satisfying homogeneous Dirichlet boundary conditions of the second order. The basis functions have small supports and wavelets have vanishing moments. We show that stiffness matrices arising from discretization of the biharmonic problem using a constructed wavelet basis have uniformly bounded condition numbers and these condition numbers are very small. We present quantitative properties of the constructed bases and we show a superiority of our construction in comparison to some other cubic spline wavelet bases satisfying boundary conditions of the same type.