Personally, I would pick the 386 myself. You have the most upgrade capabilities, it can run Windows 3.x decently well, and with a bit more $$$ down the hole you can probably get by with Windows 95, even.
Most games I see people playing on vintage systems lately are the early FPS games (Wolfenstein 3D, DooM, Heretic, Duke Nukem 3D, Quake I / II, etc), and most of those won't run at all on a 286 (or very poorly at that), let alone an 8088. Even my 20MHz 286 struggles to pull off Lotus III.
A big part of it is which machine you want to emulate. The 286 is mostly out of the running, it's very much between the 8088 and 386 in every sense. Everything the 286 can do, a 386 can do better, and the 8088 can run all that speed sensitive software some people might care about. The 286 can't.
I'd say it comes down to whether or not you need more than 640K for your software. If that's going to be an issue, you'll have to have the 386.
Oh, and the 386 will obviously be much faster for usage outside of gaming. That much should be plainly obvious. The 16-bit bus will also help you dramatically in tracking down expansion cards for things like SVGA and ethernet, and any you do find will be drastically faster than the 8088.
P.S. How much are you looking to get for the machines you're getting rid of, whichever they may be? I'd be interested in any of them.