As spacefaring rapidly expands to include more and different kinds of stakeholders, activities, benefits and risks, we need to think more broadly about what we are doing in space, especially through social, environmental and ethical lenses. What do we want from space? How should benefits and risks be distributed? What values should guide decision making, and who ultimately decides? Building on work in technology ethics and governance, this paper describes a responsible innovation framework that could help steer spacefaring in ways that are both innovative and responsible. Highlighting criteria of anticipation, reflexivity, inclusion, responsiveness and distributive fairness, the paper examines three areas where the approach could help secure a better space future for all: managing low-Earth orbit activities, space resource management and the Artemis missions. While acknowledging that responsible innovation is a highly aspirational ideal, especially in the context of rising authoritarianism and persistent geopolitical tension, the paper maintains that aspirational frameworks are essential to safe and responsible spacefaring.