We have experimentally studied the electronic $3p\leftarrow 3s$ excitation of Na atoms attached to $^3$He droplets by means of laser-induced fluorescence as well as beam depletion spectroscopy. From the similarities of the spectra (width/shift of absorption lines) with these of Na on $^4$He droplets, we conclude that sodium atoms reside in a ``dimple'' on the droplet surface and that superfluid-related effects are negligible. The experimental results are supported by Density Functional calculations at zero temperature, which confirm the surface location of sodium on $^3$He droplets, and provide a detailed description of the ``dimple'' structure. The calculated shift of the excitation spectra for the two isotopes is in good agreement with the experimental data.