I also have a 1988 3.7lx (180 hp) and I had a similar problem with the key switch as woodieman suggested. In my case, the problem happened intermittantly. In some cases I would get a no start, other times it would try and fire with starter engaged and when key was released back to the run position it would cut right out. Maybe I'm wrong about your problem but it happened to me. Do you have electronic ignition (Pertronix?). If not, I highly recommend it! The engine will idle better, you will get a little more power, better fuel efficiency, easier cold start ups etc. Some of the older setups before the upgrades have a ballast resistor as Docksidemarineservices suggests. Do you still have the original rectifier (generator) and water cooled voltager regulator setup? If so, I would get the aftermarket alternator setup. However, some of the older guys like the original points setup but I immediately noticed a positive result after installing the Pertronix electronic ignition,Flamethrower 40,000 volt coil, 8mm wires,and a regap of plugs to .040 See if you can have someone turn the key while you hold a test light to the positive side of coil where your ignition circuit is connected. If the coil losses power then you probably need a key switch. jmmho
One last thing: Does your engine have a neutral safety switch in the shift controller? Do you have a throttle only button of some sort. maybe power is being interrupted between key switch and coil by faulty neutral safety switch.
Maybe this has nothing to do with your problem also, but figured you could check to be sure. good luck, Tom