If those on the ship asked (Yonah 1:11) and Yonah answered (1:12) as far as how to save them, why did they ignore his suggestion (1:13)?
They didn't ignore his suggestion. This is how the pesukim go (according to Malbim):
1:12 — Yonah tells them that the storm is to punish him and not them, and since it's here to punish him for running away, they should throw him away so he can get his punishment.
1:13 — They reasoned that since the punishment is for running away, it would be more appropriate to return him there, but they weren't able because of the storm.
1:14 — They decided to throw him off, but first prayed to G-d not to count it as murder, since they had to throw him off to save their lives.
1:15 — They threw Yonah off.
They did not disregard Yonah's suggestion, they were trying to work out the best course of action. What could be perceived as them ignoring Yonah is actually them hesitating as they grappled with an ethical dilemma.
In Rabbi Yosef Deutsch's book Let My Nation Be Warned (Feldheim - 2014) pp.61-62, working off the mefarshim (commentators), he creates a dramatization of what occurred:
"We are honorable seamen. We don't throw people overboard. It's just not done.1 And beside, do you think the Almighty will be pleased with us if we become murderers? We cannot do such a thing. Why don't you just jump? That will solve everyone's problems."
Yonah shook his head. "You cannot commit murder, and I cannot commit suicide. I will not jump."
"Men!" he cried out. "Let's make one last try at getting back to port. We've thrown the idols overboard. We've repented. We've refused to commit murder even at the risk of our lives. Maybe the Almighty will be satisfied and bring an end to the storm."2
The oarsmen dug their oars into the water and the rest of the crew clambered over the rigging to untangle the sails. But the ship did not falter from its course westward in the fury of the storm.3 It was completely in the hands of the Almighty.4
"You are good people," said Yonah. "You don't deserve to die. I still think that you should throw me overboard, but you don't want to be murderers. And if the storm continues even after you throw me overboard you will certainly be murderers. But what if you were absolutely sure that throwing me overboard would save all your lives? Would that change your thinking?"5
"I don't know," said the captain. "It would be taking a life to save many, it would still be murder."6....
1 Kad HaKemach
2 See Rashi, Yonah 1:13
3Rav Eliezer Beaugency, Radak, Metzudas David, ibid. 1:11
4Abarbanel, Malbim, ibid.
5 According to Kad HaKemach, Yonah wished to receive suffering as penance; he was confident that Hashem wouldn't kill him.
6 Mahari Kra, Yonah 1:14