(Edited to clarify some points about fuse labelling that I had not made particularly clear. Thanks for the feedback
@SMSW.)
Check what other things do / do not come on when ignition is turned on - do the headlights work for example ?
But some quick checks first:
Make sure the gear is in neutral, the side stand is down and pull the clutch lever in. Try it starting it then.
You don't need to do this every time, but it will eliminate 3 of the things that can prevent the bike from starting.
Normally, the bike will always start if it is neutral.
It will start if it is in gear and the the clutch is held in.
It will cut if the engine is running, in gear and the sidestand is put down.
It should start in gear with the clutch held in, but not with the clutch out. Sidestand up.
You should hear the whirr of the fuel pump for a second or two when you turn the key to ignition.
If not, then there is something wrong with either the power in or to the fuel pump circuit - which also powers some ignition components. If you don't hear the fuel pump whirr when you first turn on the ignition, there is no point in even trying to hit the starter.
Check Main Fuse A - this is the 30A fuse on the starter relay to the left hand side of the battery. It is protected by a plastic/rubber sheath. It feeds the 12V supply to the ignition switch. Check the wiring and the plug-in relay in that area. Look for corrosion, damage, melting.
Don't confuse this with the Main Fuse B which sits under the brown cover on top of the battery.
There are a number of fuses in two fuse boxes behind the left hand side panel under the seat. The covers are labelled. The circuit diagram available on this forum (coloured) and the Honda manual (black and white) label these fuses with a letter.
Here are a couple of photos of my ST1300A9 fuse boxes, with the letters superimposed. The photo on the left shows the fuse box nearest to the front of the bike.
The colours indicate how the fuses get their power. Green = powered vie ignition switch.
Red = always live, protected by Main Fuse B. Magenta=Always live, not protected at all as far as I can see.
Fuses A and B are not the same as Main Fuses A and B !!
Check that Fuse D connection / blades are OK. That feeds the power to the run/stop switch.
If there is an issue with the ignition switch, you can probe fuses A B D E F in order to find out if power is getting to the fuse box from the ignition switch. These fuses are all powered when the ignition switch is on.
Then check power is coming out of the Run/Stop switch. You can do this by probing the black wire on the Bank Angle Relay. (Where and how ? Explained with photos and diagram in the linked document below).
I suspect the issue may be before the fuel cut-out circuit but just for completeness - this is a circuit which only activates if it has power from ignition, if the run switch is on, if the bank angle sensor relay is on, if the fuel relay is on and if the return to earth is enabled by the ECM (determined by other things such as neutral switch, clutch switch, side stand switch and such like. The state of the bank angle sensor also turns on and off other ignition features (like the coils), and sends a signal to the ECM which I would imagine (but don't know) that this would prevent the starter button from working.
A detailed diagram and explanation of the Fuel / Ignition Cutoff circuit in the pdf attached to the first post here. Plenty of possible culprits - but it starts with a reliable 12v feed from the ignition switch. The colours on the diagram in this document are taken from the colours in the Honda Workshop Manual.
An invaluable aid to the identification of electrical faults which result in the fuel pump not working on the ST1300. If you think that you cannot read an electrical circuit, try this. It takes you by the hand and leads you through the diagrams one step at a time. The Fuel Pump Circuit...
www.st-owners.com
Some of these switches (eg the neutral) have just one wire - so rely on the return to earth to make/brake the circuit. Earth doesn't just mean making good contact with the frame. It connection to the chassis has got to get back to the battery, so check the -ve battery lead is sound at both ends.