what does Victor's father say is a responsibility of the survivors?

The responsibility of survivors is to go on living.

‘Do you think, Victor,’ said he, ‘that I do not suffer also? No one could love a child more than I loved your brother’ tears came into his eyes as he spoke— ‘but is it not a duty to the survivors that we should refrain from augmenting their unhappiness by an appearance of immoderate grief? It is also a duty owed to yourself, for excessive sorrow prevents improvement or enjoyment, or even the discharge of daily usefulness, without which no man is fit for society.’