How to arrive from Brussels (centrum) to the Brussels airport by bus?

No, lines 12 and 21 do not go to Brussels-South, as you can see on the network map. This sketch lists all the stops and explains the relationship between line 12 and line 21.

Other ways to reach the airport are detailed on the airport's website. From Brussels-South, you could try to catch a train or find some other way to reach the European quarter and take the bus from there (e.g. from stop “Schuman”, which is also close to a train and a metro station by the same name). Note that some trains do call at other train stations in Brussels so getting off at Brussels-South might not always be the best choice if you are coming by train.

• Bus 12 does not go to the city center but to the European Quarter. If you come from the city center you can take the metro lines 1 or 5 to Schuman (<10 minutes) and change there for bus 12. The bus stop is in the Rue Archimède - Archimedesstraat. A ticket to the Airport costs €4 and is also valid for the metro. This bus doesn't ride during the weekend. Journey time from Schuman to the airport is less than half an hour (even during rush hour) and it runs very frequently (up to every 10 minutes during rush hour).
• Bus 21 is more or less the same, but stops a lot more often and consequently takes longer. During the week it doesn't go all the way to the airport, but during the weekend it is a good alternative for bus 12.
• The Flemish bus operator De Lijn operates buses 272 and 471 between Brussels-North station and the airport. Bus 471 is faster, but doesn't operate during the weekend. Tickets cost €2 if bought beforehand. Or you can buy an sms-ticket valid for 60 minutes for €1.55. Tickets of De Lijn can not be used on other operators (e.g. STIB/MIVB, the Brussels operator). This means that getting to Brussels-North will probably cost you another ticket.
• De Lijn also operates buses 359 and 659 between Roodebeek (Woluwe-Saint-Lambert) and the airport. Roodebeek is far from the city center, but easily reached using metro line 1. You will need separate tickets for the bus and the metro. In some cases this might be an option to consider.
• Not what you asked for but as @Relaxed pointed out, there is a good train connection between the city center and the airport. Train tickets to or from the airport include a special tax (Diabolo tax) making them rather expensive. Which is probably why you don't want the train. At the moment the Diabolo tax is €5.07, pushing the price of a single ticket between city and airport to €8.50.
• If your objective is to evade the Diabolo tax it might be a good option to take the train to Zaventem. This costs €2.90. From Zaventem station it is a 20-30 minute walk to the airport terminal. It is a good alternative if you don't have a lot of luggage and the weather is reasonable. De Lijn also operates very frequent buses (multiple lines) between Zaventem and the airport.