Muon collisions at multi-TeV center-of-mass energies are an ideal environment to study the properties of the Higgs boson. The high production rates of the Higgs boson at these energies enable precise measurements of its couplings to the standard model fermions and bosons. In addition, a measurement of the double-Higgs production cross section allows to determine the trilinear self-coupling of the Higgs boson and thus to probe the Higgs field potential. This contribution presents the expected statistical sensitivities on the Higgs boson production cross sections at a 3 TeV muon collider with a dataset of 1 ab$^{-1}$. The results are obtained using a detailed detector simulation for the signal and physics background samples and take into account the effects of the beam-induced background on the detector performance.