The physical features exhibited by Hermite-Gaussian (HG) beams propagating in nonlocal nonlinear media with Gaussian-shaped response are discussed with an approximate variational method. Using direct numerical simulations,we find that the beam properties in the normalized system are different with the change of the degree of nonlocality. It is shown that initial HG profiles break up into several individual beams with propagation when the degree of nonlocality α is small. HG beams can propagate stably when α is large enough.