Rookies are filler. They contribute very little but you are penalized if the crew bar isn't full.
Experts provide skill to your crewman.
Veterans provide 4x the skill that Experts provide.
There is a maximum skill level for each kind of sailor, but it's very hard to hit until after level 100. Until then, investing in as many veterans as you reasonably can is wise, especially for crew like Fighter pilots, Seamen, and Engineers.
The maximum amount of veterans a sailor can hold equals 40% of his crew size. So if a sailor has 300 crew on it, he can have 120 veterans.
Boosting (Premium Sailor) provides a 20% bonus to skill. It's a simple multiplier so it scales with level. Make sure you put lots of vets and experts on the sailor though, because 120% of "not much skill" still doesn't do a lot. The more vets and experts your sailor has, the greater the effect of adding 20% becomes.