This may be a good question for the Distributions...
What distro are you trying to install on the 386. Good luck trying Mandrake or even Redhat. (I don't want to hear from the redhaters on 386's) My experience with both those distros where they were very heavy on the apps (disk space) and geared for mostly for X support.
I have had slackware running on a 386 (over 6 years ago) and more recently (~2 years ago) on a 486 DX2. Slackware is a good install if you don't need any of the fancy stuff and you can really trim down your installation size. Unlike Mandrake, where I tried twice to NOT install X and it still stuck it on the drive.
There are a few good distros that will work well on a 386, you just have to try them out. There have been a couple discussions on small installs on this board, try a search.
Also try
http://www.linux.org and go to distributions.
Gary