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In the question "Can I safely wait in the transit area of a destination airport for my visa to be approved?", OP explores the idea of traveling on an electronic visa which is not yet issued.

This rather risky venture got me curious. The unanimous consensus here seems to be that the airline will check OP's visa status and thus deny him boarding. I certainly do not mean to dispute this assertion, this is a question of legitimate curiosity.

How exactly do airlines do this? There are dozens of governments in the world that issue electronic visas. Do airlines have computer systems that interface with the internal databases of these governments, to check whether each traveler has a valid electronic visa at any given time?

Or do the airlines simply demand that the traveler show a print of the email they got when the electronic visa was approved, or something like that? Needless to say, such an email is rather easy to forge if no electronic check is made.

In "Do airlines verify the validity of visas electronically during check-in?", the answer seems to be "no", but that is for normal visas, that are printed on paper, with all the security features that would offer.

How do airlines check whether a traveler has an electronic visa?

I have tried searching the web in order to answer this myself, but the large number of irrelevant results has been too large for me to trawl my way to the answer.

Airlines do have computer systems that do this, yes.

Many countries, such as the US, require airlines to transit information about inbound passengers prior to flight departure through an Advance Passenger Information System (APIS). Brazil, which the OP mentioned, is one such country. Brazil, along with a number of countries, uses a product called SITA iBorders to take care of this on the government's side; one module handles e-visa issuance and verification:

Supports denial-of-boarding checks that allow a government to authorize travel in real time, one passenger at a time, at flight check-in. Data known in the industry as
interactive API, or advance passenger processing (APP) is exchanged between airlines and the iBorders system. Denial of boarding checks may include final watch list and risk assessment checks as well as validation of travel documents against the ETA, visa and passport and other databases. The system supports denial-of-boarding checks for 11 governments, with more than 200 million passengers processed annually on behalf of 130+ airlines.

Other countries may have their own systems or use different vendors (such as ARINC). The airlines, in turn, connect their IT systems to vendors that know how to talk to these systems. Airlines need permission from a country to operate flights there, with many conditions imposed as part of that process, so complying with immigration and data interchange requirements is just one of the many tasks they need to address before they start service.

Not every country has an APIS system, and it's possible that there are various gaps in the system (I have no idea if Azerbaijan e-visas are really electronically checked before boarding, for instance, though they do seem to encourage printing them out and talk up how the printout has a machine readable zone, which makes me suspect not), but since airlines can receive substantial fines for transporting passengers without proper documentation, it's mandatory for many of the largest travel markets.

Yes, each airline has a system to check any E-Visa and if a visa is required for a passenger. I work for Delta Airlines and here is their system to check:

https://skyteam.traveldoc.aero