Stars collect most of their mass during the protostellar stage, yet the accretion luminosity and stellar parameters, which are needed to compute the mass accretion rate, are poorly constrained for the youngest sources. The aim of this work is to fill this gap, computing the stellar properties and the accretion rates for a large sample of Class I protostars located in nearby (<500 pc) star-forming regions and analyzing their interplay. We used a self-consistent method to provide accretion and stellar parameters by modeling the spectral energy distribution and using veiling information from near-IR observations when possible. We calculated accretion and stellar properties for the first time for 50 young stars. We focused our analysis on the 39 confirmed protostars, finding that their mass accretion rate varies between ~10-8 and ~10-4 M ⊙ yr-1 in a stellar mass range between ~0.1 and 3 M ⊙. We find systematically larger mass accretion rates for our Class I sample than for Class II objects. Although the mass accretion rate we found is high, it still suggests that either stars collect most of their mass before the Class I stage, or eruptive accretion is needed during the overall protostellar phase. Indeed, our results suggest that for a large number of protostars the disk can be unstable, which can result in accretion bursts and disk fragmentation in the past or in the future.