How do you know if boat ignition switch is bad?

Not a valid assumption... ignition switch could be defective.

To check which is bad...

There are several places on the solenoid of interest;.

There are heavy wires... one long to the battery, the other short to the starter motor itself.

There are thin wires ( one or two)... If two, one goes to ground. The other is the lead from the ignition switch.

If you jumper from the battery to the heavy lead, and it starts (cranks) then the started motor itself is OK and you could have EITHER a bad solenoid or a bad ignition switch.

If you jumper from the battery to the thin wire from the ignition switch and it cranks, then its a bad ignition switch ( or wire to it). If it does not crank...bad solenoid.

If it does crank when you jumper the thin wire, then get a short piece of wire and climb under the dashboard. At the back of the ignition switch connect the wire between the heavy wire (often purple or red) and the terminal on the ignition switch with only one wire on it. If it cranks...bad switch...If it doesn't crank...wiring problem. If both of the other switch terminals have only one wire (rare), then the one to jumper is the one furtherst from the heavy wire IN THE DIRECTION OF ROTATION of the switch as you go from off to on to start."