The worldwide spread of a new coronavirus (SARS-CoV-2) since December 2019 has posed a severe threat to individuals' well-being. While the world at large is waiting that the released vaccines immunize most citizens, public health experts suggest that, in the meantime, it is only through behavior change that the spread of COVID-19 can be controlled. Importantly, the required behaviors are aimed not only at safeguarding one's own health. Instead, individuals are asked to adapt their behaviors to protect the community at large. This raises the question of which social concerns and moral principles make people willing to do so. We considered in 23 countries (N = 6948) individuals' willingness to engage in prescribed and discretionary behaviors, as well as country-level and individual-level factors that might drive such behavioral intentions. Results from multilevel multiple regressions, with country as the nesting variable, showed that publicized number of infections were not significantly related to individual intentions to comply with the prescribed measures and intentions to engage in discretionary prosocial behaviors. Instead, psychological differences in terms of trust in government, citizens, and in particular toward science predicted individuals' behavioral intentions across countries. The more people endorsed moral principles of fairness and care (vs. loyalty and authority), the more they were inclined to report trust in science, which, in turn, statistically predicted prescribed and discretionary behavioral intentions. Results have implications for the type of intervention and public communication strategies that should be most effective to induce the behavioral changes that are needed to control the COVID-19 outbreak.