Is it permissible to use money inherited from one's father if its source was usurious banks, [received] in good faith?

If a father deposits his money in interest-based banks solely for safekeeping without earning interest, this does not affect your ownership of his inherited wealth. The prohibition here is from the perspective of not cooperating in sin. However, if he deposits it with usurious interest, then these interest earnings are not permissible for you. You must disburse them and spend them on general Muslim welfare. You have the right to inherit the remainder of his wealth, as Allah Almighty says: "And if you repent, you may have your principal – [thus] you do no wrong, nor are you wronged."