The word “abide” [Greek: menō] invariably carries the idea of staying put or remaining in a relationship or condition. Romans 6 teaches it is contrary to a saint’s calling and nature to remain in sin. Our “old man” (the person we once were) was crucified with Christ. We may sin, but sin has no longer has any legal right or claim to keep us enslaved.
The opening question — Are we to continue in sin that grace may abound? — is what one might expect to hear from a lawyer at a trial. Paul has presented his case; believers are justified and have peace with God; a glorious future is ahead.