How do you run .exe files with wine by clicking on them?

In the Finder, select one of your .exe files.

Finder -> File -> Get Info

Open with: ...

Click on the pop-up list and choose Wine if it is in the list, or Other and navigate to Wine.

Once Wine is selected to open for that .exe, click on "Change All", so all .exe files will open with Wine.

So wrap Wine in an application, and bind the .exe's to the wrapper application.

Platypus (free download)
<http://www.versiontracker.com/dyn/moreinfo/macosx/19870>
Can be used to turn any Unix program or script into a double clickable application, as well as a drag and drop target application.

It should be possible to wrap Wine with Platypus.

As an alternative, you might try Applications -> Automator -> Run Shell Script workflow, and save that as an application. Then bind your .exe's to that workflow you created.

NOTE: I have not played with Wine, so your mileage may vary.

Platypus worked well for me for this problem, with one "gotcha."
I set up the Wine app and tried to open an EXE with it to no avail.
It turns out that version 4.2 (current as of this writing) has a bug and does not pass in the parameters properly.

The author promises to fix it in 4.3 and recommends using 4.0 until then if you need this feature.
http://www.sveinbjorn.org/platypus

Setting things up in 4.0 worked like a charm!

(Hopefully this message will be obsolete soon...)