We propose that the finite size of the Kondo screening cloud, xi(K), can be probed by measuring the charge quantization in a one-dimensional system coupled to a small quantum dot. When the chemical potential mu in the system is varied at zero temperature, one should observe charge steps whose locations are at values of mu that are controlled by the Kondo effect when the system size L is comparable to xi(K). We show that, if the standard Kondo model is used, the ratio between the widths of the Coulomb blockade valleys with odd or even number of electrons is a universal scaling function of xi(K)/L. If we take into account electron-electron interactions in a single-channel wire, this ratio also depends on the parameters of the effective Luttinger model; in addition, the scaling is weakly violated by a marginal bulk interaction. For the geometry of a quantum dot embedded in a ring, we show that the dependence of the charge steps on a magnetic flux through the ring is controlled by the size of the Kondo screening cloud.