You will need a Blizzard Battle.net account.
With a Battle.net account, you can add game keys for Blizzard games you own, including StarCraft. Old CD keys will be converted to newer, 26-character keys needed for the Anthology Edition. You can also purchase the game online if you don't own a CD copy.
Once your key is registered or purchased, "StarCraft Anthology" will be listed in the "Classic Games" section of your account. Following the link will lead to the downloader page. If you're running a Mac, choose the Mac downloader. If you're running Linux, the Windows downloader may work in Crossover.
Run the Blizzard downloader to download the game installer. Expect a download time of about 30 minutes. Save the installer to a location on your computer that you can find. The installer package contains both the Mac and Windows installers; it does not matter if you use the Mac or Windows downloader application to retrieve it.
When the downloader is complete, it will ask if you want to install StarCraft. Decline this because you want to use Crossover for the game installation.
Start Crossover and install StarCraft as any other application using the installer package you downloaded. I used a Windows 2000 bottle. A window will appear asking for a location within the bottle to install StarCraft. C:\Program Files\ is your likely choice.
Once installed, test the game. Using Crossover 11.2 on Mac OS X 10.8, the game runs full-screen for some but has problems for others. The game will run in a Virtual Desktop window without problems but resolution is limited to 800 x 600.
Connecting to multiplayer Battle.net experiences problems. The automatic updater does not work. See separate page on Codeweavers.com for installing the 16.1 patch. Note that the Anthology Edition contains both the original StarCraft and the Brood War expansion, so you will want to apply the Brood War patch.