In TaskBullet Settings → Basecamp Assign, click Connect Basecamp. You will see Basecamp / 37signals Launchpad and authorize TaskBullet. We use OAuth — we never see your Basecamp password.
Only an admin of your company's own Basecamp should connect. If your work already lives in a project on TaskBullet's Basecamp, leave that project as it is. Do not connect TaskBullet's account. If you are not an admin of your company's Basecamp, ask that admin to connect it.
After you approve access you return to TaskBullet Settings. Connecting alone does not put TaskBullet under Assign in Basecamp — finish step 4 here first. If you stay in Basecamp looking for TaskBullet, you will not find it yet.