It must be "눈을 감은채 살아도 좋을까?", which means, "would it be all right to live with my eyes closed?" or "Would it be all right to live, keeping my eyes closed?" -은채 shows the progressive state of verbs. 눈을 감다 means to close eyes. 눈을 감은채 means "with eyes closed." Actually, 감은채로 is the more proper form. And the form changes depending on the pronunciation of verbs. I don't know the grammar rule for this, though. I believe another native speaker could give you a better answer !

need more context to answer the question properly