I had the same problem. 91 honda accord Lx manual 200k miles wouldn't crank at all. Figured it had to be either battery, starter, solenoid, bad ground, blown fuse, bad starter fuse, safety clutch switch (the top one on the clutch pedal), or bad ignition switch in the start III position.

battery checked out.
starter checked out.
voltometer reads no power to the remote wire on the solenoid.
next, test starter switch. Jump the plug with a paperclip.
Does car start now?
Okay the switch is bad. edit: "starter interlock switch" as someone else referred to it.

if the switch paperclip (jumping the switch) doesn't work, try the ignition switch next.

That's how i solved the same problem. Hope it works out for you.

BTW the switch is hard to get to. it is under the dash up high buried behind a bunch of wire harnesses at the near top of the clutch pedal.