We introduce the public version of the BAyesian STellar Algorithm (BASTA), an open-source code written in PYTHON to determine stellar properties based on a set of astrophysical observables. BASTA has been specifically designed to robustly combine large data sets that include asteroseismology, spectroscopy, photometry, and astrometry. We describe the large number of asteroseismic observations that can be fit by the code and how these can be combined with atmospheric properties (as well as parallaxes and apparent magnitudes), making it the most complete analysis pipeline available for oscillating main-sequence, subgiant, and red giant stars. BASTA relies on a set of pre-built stellar isochrones or a custom-designed library of stellar tracks, which can be further refined using our interpolation method (both along and across stellar tracks or isochrones). We perform recovery tests with simulated data that reveal levels of accuracy at the few percent level for radii, masses, and ages when individual oscillation frequencies are considered, and show that asteroseismic ages with statistical uncertainties below 10 per cent are within reach if our stellar models are reliable representations of stars. BASTA is extensively documented and includes a suite of examples to support easy adoption and further development by new users.