Actions on climate change which are not supported by all countries are not very effective. However, full participation in a global climate treaty with meaningful emission reductions is difficult to achieve. The non-excludability of the public good mitigation provides an incentive to abstain from global action. Moreover, carbon leakage renders it unattractive to join a treaty without full participation. We study whether and under which conditions border carbon adjustments (BCAs) can mitigate free-riding and reduce carbon leakage in a simple strategic trade model. We show that BCAs can lead to large stable climate agreements, including full participation, associated with large global welfare gains if treaties do not restrict membership (open membership), as this is typical for environmental agreements. We caution against restricting accession to treaties (exclusive membership), as this is typical for trade agreements, which may serve individual but not global interests.