I've seen it running well with System 7.6.1, Open Transport 1.1.2 and FreePPP, which is remarkable for a piece of software from 1993. However, I've yet to get it working with System 6 on my SE. It can be done, or so I'm told. Exactly what combination of software is required I've yet to discover, and this page is very much an ongoing project.
If you want to experiment for yourself, you can download the browser and MacTCP 1.1.1, which may be needed:
Download MacWWW 1.0.3 and MacTCP 1.1.1
Download As above, but also includes later versions of MacWWW
Here's an extract from Matti Haveri's "68000.txt" FAQ:
"MacWWW 1.03 (Samba) works on 68000 Macs (at least on System 6.0.5 - it crashes on System 7.0.1*). It opens fine, loads the startup page fine (doesn't know how to read files from disk, text-only, no ISO 8859-1 translation, opens a new window for each link).
"Do not close any windows as this crashes the Mac; also quitting MacWWW seems to cause a crash. Note that with System 6, MacTCP may crash when saving prefs for the first time, so check that the prefs are actually saved".