Accurate as 15 Feb 2017
By train or bus. Brussels Airport has their own underground train & bus station.
You can look for public transport possibilities on the website of the national traincompany (NMBS in flemish, SNCB in french) or on the website of the Flemish busses(De Lijn). Both sites should give the same results (both give trains and busses)
I can't give you a link to results because that search query expires.
Fill in the route planner as in the example below. (Luchthaven means airport)
On weekdays there seems to be 1 train/hour, which takes 4 minutes. This would cost you €7.50 one way. (prices do change almost every year though)
You can also take the bus from Zaventem station to the airport. This will take between 15 - 20 minutes, depending on which bus. (This is also a week day result)
You can use the website from the busses company
Prices may change and were from 16/02/2017
Busses are a bit cheaper. There are serveral tickets you can buy. Thinking you will only take the bus once:
1 way ticket: €3 easiest, can buy on the bus(cash), from a machine or in the bus station. (sometimes you can buy it from the train station people too)
SMS ticket: €2 + €0.15 (for the sms itself), cheaper than 1 way ticket. Valid for 1 hour from the moment you send your SMS. (depends on what you put in the message). Only works with these mobile operators
Both are only valid for 60 minutes.