The effect of a hard wall on the clustering and continuum percolation of a hard spheres fluid is studied using Monte Carlo simulations and connectedness theory. We calculate an averaged pair-connectedness function rho dagger(r;z) which is the probability density of finding two particles in the same cluster and separate by a distance r under the assumption that one of them is fixed at a distance z from the wall. We also obtain the mean size S for the cluster containing the fixed sphere and the critical percolation density rho(c) at which it becomes macroscopically large. Monte Carlo results allow us to conclude that, for given number density and connectedness distance, the wall causes the decrease of S and the increase of rho(c) in comparison with those found for the bulk in the absence of the wall. Both effects diminish with increasing z. The simulation data also show that, in the presence of the wall, the clusters are eccentric with cylindrical symmetry, slightly flattened in the region of contact with the wall. The theoretical calculations involve the solution for rho dagger(r;z) of an integral equation. It is derived from the one proposed some time ago by Giaquinta and Parrinello to obtain the average of the ordinary pair correlation function in the presence of the hard wall [J. Chem. Phys. 78, 1946 (1983)]. Integrating the pair-connectedness function over r we have S whose divergence determines the theoretical critical density. The results so obtained are in satisfactory agreement with Monte Carlo data.