How do I enable font anti-aliasing in Wine?

The easiest way is through winetricks

mkdir ~/bin
cd ~/bin
wget http://winetricks.org/winetricks
chmod +x ./winetricks

winetricks fontsmooth-rgb

In newer version of winetricks the command to use is

winetricks settings fontsmooth=rgb

There are other LCD pixel orders available. You can read about them and the rest of the impressive winetricks feature list on its WineHQ site.

Just adding details to Oli's answer for those who need it: what winetricks settings fontsmooth=rgb actually does is creating and importing a registry file with the following content:

REGEDIT4

[HKEY_CURRENT_USER\Control Panel\Desktop]
"FontSmoothing"="2"
"FontSmoothingGamma"=dword:00000578
"FontSmoothingOrientation"=dword:00000001
"FontSmoothingType"=dword:00000002

Once you saved it as fontsmoothing.reg you can import it with:

wine regedit fontsmoothing.reg