We study the number of connected spanning subgraphs f_d,b(n) on the generalized Sierpinski gasket SG_d,b(n) at stage n with dimension d equal to two, three and four for b=2, and layer b equal to three and four for d=2. The upper and lower bounds for the asymptotic growth constant, defined as z_SGd,b=lim_{v →∞} ln f_d,b(n)/v where v is the number of vertices, on SG_2,b(n) with b=2,3,4 are derived in terms of the results at a certain stage. The numerical values of z_SGd,b are obtained.