Skooter is right...there is a fuse. If you have an owner's manual, there will be a wiring diagram in the back pages. If you don't have one, you can download one for free on the Honda Marine website.
My experience with problems like this is that it USUALLY turns out to be a bad connection SOMEWHERE.
Remember...you can have PLENTY of volts BUT...if you have TOO MUCH RESISTANCE in a circuit (bad/loose/dirty connection)....no CURRENT will flow. Electrical class instructors equate it to a garden hose...
...you can have 80 psi water pressure but if you have a bad kink in the hose.....no water will come out the end. Electricity works pretty much the same way.
If that outboard has it's own battery for cranking...start there. REMOVE and clean BOTH the positive AND the negative clamps. Move along BOTH the cables and clean the connections at the end of EACH ONE. Most people focus on the POSITIVE cable and it's connections but OVER 50% of problems like this are due to a poor NEGATIVE connection.
Yes...of course...check the fuse. Neutral safety switch...SURE! If you bypass it and the engine cranks...there's your culprit.
Most of these outboards also use a device called the "main relay". It typically houses an operational relay for the starter solenoid. There MIGHT be a problem there but THAT is expensive and you would NEVER want to change it unless you had gone over everything else first. There is a testing procedure for determining if the main relay is bad.....but it's NOWHERE near time for that now.
There is also a remote chance that one of the big cables has "gone south". They can deteriorate UNDERNEATH the insulation and you can't see it but a thing we call "green death" takes place. It is Copper SULPHATE (oxidation). A result of water and air acting to oxidize or...rust...if you will, the many wires that make up a large battery cable. The only way to find a bad cable like that, without removing the insulation and having a look, is to perform what is known as a "voltage drop" test. If it comes down to that....you can Google it and learn how to do it. It's easy IF you have a RELIABLE and ACCURATE electrical multi meter.
But my money is on THE BAD CONNECTION. It takes a bit of "digging" to get to the ends of the cables and a bit of elbow grease to shine everything up like a NEW PENNY but it will LIKELY yield some good results.
Good luck.