I got this running on OSX, I can give you two versions of the solution, one that's fairly easy and basically works, and another that's more advanced, but the game also runs more smoothly.
For reference I did this also using Crossover Games on Snow Leopard on a Macbook that's a few years old, so if you've got a fast Mac, you might be able to do the easy solution and have it run fine.
If you're not using Crossover, presumably you can use Wine to run the installer. I am assuming that you have it currently installed, but it runs with just a black screen in Crossover/Wine. I'm using Crossover Games 8.1.4 which is old and some compatibilities may be better or worse for you.
Easy Solution
Using Crossover, choose Programs->GOG->LIttle Big Adventure->Graphic Mode Setup.
You're going to get this funky window that only sort of half works. (You can't click on anything). Press the TAB key several times until you highlight the "Graphic Mode" pulldown. Use your arrow keys to choose anything that's not Overlay (try Surface). Then TAB a few more times to Save Settings, hit enter. TAB some more, then hit start game. Voila, it should load up. After you've done this once, you should be able to just launch the game normally.
Now the thing to realize is that this solution is essentially going through two layers of emulation (Mac emulating Windows, Windows emulating DOS). We can cut out the middle man and have the Mac directly emulate DOS. This should run more smoothly (and it does in my experience) which leads me to...
Advanced Solution
(Before I begin, can I just say that I'm surprised it's so difficult to make an icon on Mac OSX that runs some program with command line arguments?! This would take 2 seconds on any other platform.)
First download and install DOSBox for Mac in your applications folder.
Next, open the AppleScript Editor (you can use the Mac spotlight magnifying glass in the top right of the screen to find it quickly). Now you're going to put this code into an empty script:
do shell script "cd '~/Library/Application Support/CrossOver Games/Bottles/<LBA BOTTLE NAME>/drive_c/Program Files/GOG.com/Little Big Adventure'; /Applications/DOSBox.app/Contents/MacOS/DOSBox -conf ./dosboxLBA.conf &"
ATTENTION: replace <LBA BOTTLE NAME> with the name of the bottle that you installed Little Big Adventure into.
(Another aside, just so I can explain what the script does. do shell script is some AppleScript way of saying "do this." The first thing we tell it to do is to go into the directory where Little Big Adventure is installed (cd '<name of directory>'). Then the semicolon is the start of the next command, we tell DOSBox to run with the command line argument -conf ./dosboxLBA.conf, which just tells DOSBox to look at that file for configuration settings (it also contains the commands that run the game). The & just tells the script engine not to wait for the command to finish.)
Now save the script as an application (That's important). You can save it to your Applications folder or Desktop or whatever you want. Now if all went well you should be able to double click that and it will launch the game. I would still recommend changing the Graphics Mode to Surface as in the Easy Solution, it seemed to be the best for me.
You can even edit the icon of your new app and put it in your Dock.
Have fun!
Post edited October 16, 2011 by achoo5000