For Hammer of Thyrion:
Go to the options menu and set both "mouse look" and "use mouse" to on.

If you're not using Hammer of Thyrion:
To enable the "mouse look" option seen in the source port's settings menu, you can type "+mlook" in the developer console; however, you would need to do it every time you launch the game, which is obviously not ideal. Instead, you'll want to have the game activate that command automatically, and to do that, you need to modify the autoexec options.

To do so, locate the "Autoexec.cfg" file found in the Steam/steamapps/common/Hexen 2/data1 folder and open it with any text editor; it should be empty by default. Type +mlook in the first line and save the modified file. This is what Hammer of Thyrion's "mouse look" option does, essentially.

At this point, your mouse input should be detected, but it won't work as you'd expect it to in-game. Depending on the "mouse look" setting, you'll be able to look left and right or up and down, but not both. That's because the mouse is also used for strafing, which is a ridiculous control scheme that can't be disabled in the menu for some reason (at least in Hammer of Thyrion). To fix this, you'll need to use a simple console command.

To bring up the developer console (and to exit it afterwards), simply press the ~ (tilde) key in-game. Once there, type lookstrafe 0 and confirm by pressing enter.

Now the mouse strafing should be disabled and you should be able to look in all directions as expected, except you'll come across one more issue: when you walk, your view will abruptly center itself all the time. To fix this, you need another console command: lookspring 0. After entering and confirming it, exit the console and your controls should now be working correctly. Unlike "+mlook", these two commands do remain active after you restart the game.