Good question. His profile on his(?)
website shows no recent posts in over a year. Maybe he's on a break. Can't blame him -- look at some of the work he's done.
FWIW the lightest FTP program that actually is nice is Fetch 2.1.2, I"m a big fan of it, but it's also about 500KiB on disk and uses about the same in memory. The trusty old Info-Mac archives may have something?
https://web.archive.org/web/20160418233005/http://archive.info-mac.org/
I doubt any of that is meant for super thin resource requirements. Probably shareware/freeware, I don't think they had much else up there. AFAIK most people had Pluses if they were fooling around with TCP stuff back in the day, or better, if they had a 128K/512K they were on BBS using a terminal emulator.
Just thinking about the FTP program, Font/DA Mover .... *walks over to the SE/30*...hold on, I need a memory utility to find out how much it uses, will update shortly... Anyways is pretty lightweight and can show directories/copy/move things over, I can't imagine that being that far off from the minimum you need to browse/navigate/PUT/GET files remotely.
EDIT2: Right here we go. There's two programs I know of (I hope there's a few more) that can show memory usage. MemINIT 2.0.5, which can be found readily on umich's public software source (I'm aware that 68kmla is not a file sharing source, but it's freeware), and Memory Mapper. The former...says it needs at least System 5.0 (S4.2/4.3, F6.0) and a 512Ke. It works by offering a pie chart which overlays the Apple menu, and/or drawing a 1px line over the screen image which is used like a line graph. I don't think this will screenshot properly because it's hooking into the Vertical Blanking task. So below are two images: one showing the CDEV itself, showing the settings, and the other with Font/DA Mover open.
Each tic mark denotes 64KiB. There is one white 1px mark right above the cursor. Based on that we can judge that it's using roughly 96KiB, maybe 100KiB. That's going to be pretty tight on a 128K, but not impossible (I'm accounting for the TCP and FTP routines). Then you also need space to copy files in blocks in memory, I think.