Our Kenmore 665 dishwasher has run reliably for years. Recently, water started to come out of the unit rather than draining while it is washing.

Where is the water coming out? If it in the front, the gasket could be hard and need replacement or there is a piece of debris in the gasket sealing edge. Look here first. If it is coming from under the washer, your problem is more serious and with your comment... "reliable for years", could be the seal on the bottom is leaking or the tub rusted out. If that's the case, probably not worth having it fixed and time for a new one. 10-15 years is a very good time for a DW. .

Some other things that may cause the dishwasher to leak is:
1. A split or hole in one of the spray arms that cause "waves" which cause leaks.
2. Make sure it's not sudsing. This will cause leaks in the dishwasher also. If there's suds, then someone put the wrong cleaner in it or the rinse aid dispenser is leaking.
Rich
Western Appliance Repair Boise

Have you tried checking the float device in the right hand corner at the front of the tub. I believe if that float is stuck in the down position the machine does not get a signal to shut off the water when it is washing the dishes.