Your crew is imperative to many ship functions. Without crew your ship will not fire and you cannot fly.
Your crew is a major factor in determining your damage received and your damage taken. How large this factor is, is influenced by you and your opponent's strength stats. Basically, if you and your opponent have the same crew, with the same strength, and the same weapons, you'll be hitting the same damage. If you have less crew than your opponent, they gain an increase in damage, and your damage will be a bit lower.