The airline wants to avoid a situation where they fly you to another country and the immigration officials decide you can't be admitted to the country. In such a situation, the airline is usually liable for flying you back to your point of origin at their own expense, and in some circumstances they can be fined as well.
This means that you should always be sure to provide the airline with the information about the passport that will most easily "let you into" the country you're traveling to. For example, suppose you were traveling between South Africa and the Netherlands. South Africa allows visa-free entry to Dutch citizens for 90 days; but South African citizens must have a visa to enter the Schengen area. Since the Dutch passport would allow you to enter both countries, it would make things easier to provide the airline with your Dutch passport information in both directions.
Of course, there could arise a situation where none of your passports allows you entry to all of the countries you're traveling to. If you try to board a flight and the airline does not think you'll be permitted to enter the destination country, they'll deny you boarding at your point of origin rather than risk incurring the expense of having you turned back at the border. In such a situation, when you arrive at the airport, you can provide the airline with supplemental information (a visa or a second passport) to prove that you're likely to be admitted to the country of destination. In this case, the passport with which you booked the ticket becomes immaterial.