If you dig into the river from the same Z-level, water pours out in a reasonable manner and you probably won't lose your miner.
If you dig in from underneath by digging an upward ramp, water will pour out like crazy and you might lose your miner.
Also you generally have a z-level with water in it and a z-level above it where there is no water but it's an open space. So if you expose that part, you won't get any flooding and it will map out the cave river for you. But you also won't get a notice of wet walls.
I recommend digging out a shaft going down a ways off from where you think the river will be. Dig down to the bottom. Branching off dig 2-wide tunnels down the middle of the likely embark map square, because features tend to be seated in the middle of a square. Build doors at the entrance to the staircase shaft, so if you do get flooding you can just lock the doors and you won't fill up your basement.
Also, explore only on every other level. You're wasting your time exploring the other half of them, since you'll either break through on the river's surface level or get a wet wall warning by exploring every other level.
Once you figure out where it is, turn off mining on all your good miners, set up your floodgates and link them, and have a peasant go mine out the last wall. That way, if you lose the miner it's a peasant you don't care about.
This is also a good opportunity to get rid of all the peasants by drowning. It's faster than starvation and madness.