How should we treat people who are so seriously ill or injured that there is no hope of recovery?
Literal
Illness [or-や] injury [possessive-の] degree [subject-が] very serious, recovery [possessive-の] hope [subject-が] not-exist person [object-を] how treat [should-ばいいのでしょうか].
A complex sentence with nested relative clauses: 程度が非常に重く modifies 人, and 回復の見込みがない also modifies 人. ばいいのでしょうか is a polite way to ask 'what should one do' — ば conditional + いい + のでしょうか (polite wondering). Addresses a serious ethical question.