Not with DOSBox I fear. DOSBox is made for DOS games, not Windows games. It can run some Windows 3.1 games with a copy of Windows 3.1 and some tweaking, but it's simply not the tool for running Windows 95 or newer games.

Most Windows 95 and newer Windows games run well enough in Windows XP. I'm not a Mac guy but can't you run XP without too many problems on new Macs?

You could use Boot Camp to install an XP partition, or go with a virtual machine like VMWare Fusion or Parallels.

There is a version of Wine available for Macs. Called Darwine.
I personally have never used it so I can't say anything about how easy it is to use, what it requires, how stable it is, what games it runs, etc...
The official (and rather old version-as in still offering builds for the PowerPC) is at
http://darwine.sourceforge.net/
Newer, but unofficial builds are at
http://www.kronenberg.org/darwine/
Again I've never used it, so I can't really anything about it. I'm just throwing it out as a possibility.

American McGee's Alice is unlikely to run on DosBOX because it needs DirectX. I suggest using Parallels or Bootcamp for that.

-Oscar