A search is presented for $\tau$ slepton pairs produced in proton-proton collisions at a center-of-mass energy of 13 TeV. The search is carried out in events containing two $\tau$ leptons in the final state, on the assumption that each $\tau$ slepton decays primarily to a $\tau$ lepton and a neutralino. Events are considered in which each $\tau$ lepton decays to one or more hadrons and a neutrino, or in which one of the $\tau$ leptons decays instead to an electron or a muon and two neutrinos. The data, collected with the CMS detector in 2016 and 2017, correspond to an integrated luminosity of 77.2 fb$^{-1}$. The observed data are consistent with the standard model background expectation. The results are used to set 95% confidence level upper limits on the cross section for $\tau$ slepton pair production in various models for $\tau$ slepton masses between 90 and 200 GeV and neutralino masses of 1, 10, and 20 GeV. In the case of purely left-handed $\tau$ slepton production and decay to a $\tau$ lepton and a neutralino with a mass of 1 GeV, the strongest limit is obtained for a $\tau$ slepton mass of 125 GeV at a factor of 1.14 larger than the theoretical cross section.