I have a 1995 1500 Goldwing SE. I went to start the bike, when I hit the starter, bike started, but the starter kept running. I turned the key off, but starter kept running. I had to unhook the battery for the starter to stop. With the key off, I re-connected the battery, and the starter started running.

try replacing the starter switch see if that does it

starter button on handlebar could be bad or stuck.

It may be a faulty starter solenoid ( or magnetic starter switch, whichever your bike has). If it seems to start fine when cold, but not when warm, you'll need to locate where that switch/solenoid is ( on my Goldwing, it is mounted on the side of the battery box).
Follow the positive lead from the battery - it will go to the sol/switch.The switch will have 2 larger red wires - one from the battery and the other from the starter. When the bike won't start, you can try bypassing the switch by using 2 small flat bladed screwdrivers. Placing the tip of each screwdriver on the nut side of the switch ( not on the threads) with the key on, touch the shafts of the screwdriver together - you may need an assistant to hold the clutch handle in to complete the circuit. If it starts, you've found the problem. If not, you may have a faulty starter.
Hope this helps and thanks for using Fixya!!

you may have blown the starter solenoid or even worst the starter motor
i cannot diagnose without seeing the bike properly
i strongly suggest you put in a workshop and get it diagnosed properly

check and make sure u are getting fuel to the carb...could be the fuel filter is stopped up

it the solenoid i had the same problem

The fuse on the starter solenoid is the main fuse for the bike. If that fuse was removed and replaced with a wire, you could melt down the entire harness or even burn up the cycle. The short that was popping the fuse has probably melted part of the harness or popped fuses in another part of the bike. Start checking in the main fuse box in the fake tank for shorts. Also check the solenoid with the main fuse for melt down, good luck.

How is your battery? Sure sounds like it's too weak to turn over the starter.

Assuming that it always started just fine and then the battery is the problem. Many times it slows down gradually--like it turns over slower and slower until the battery finally has no power.

Try charging it.

If you jump the solenoid and it doesn't turn over, it is probably a bad starter. But don't forget to make sure the neutral light is on and the kickstand switch is good.