Did you try the things I suggested? They are the most common reasons for problems.
Unpack trainer into the Steam YSC game folder where the game's EXE file is located; make sure trainer version matches game's version exactly; disable/turn off all security, anti-virus (especially real time scanning) and firewall, right-click on trainer and game and run AS Admin.; Press F1 at main menu. See info text that comes with trainer for any special instructions.
All four currently released Ys games were updated over the last two days so that's probably why the trainer(s) don't currently work.
From what I read on the Steam forums there wasn't that much info available but the developers have updated the cloud saving feature that can be used with Steam to synch data so that it'll work more like their future releases.