Your best bet at taming one of these is to build a small pit-fall nearby and lead it inside (Baryonyx are very fast, so make sure you can outrun it. Otherwise, a Pteranodon may do the trick.) Use a crossbow and tranq arrows to knock it out. They are pescatarians, so make sure you have plenty of fish meat to tame them, but keep in mind that this will take much longer than other methods. It is highly recommended to use their prefered kibble, however it may not be readily available. Prime fish also does the trick pretty well, which is relatively easy to farm off of Coelacanth and Sabertooth Salmon with the metal sickle.

Once tamed, a Baryonyx will absolutely change the way you play the early/mid-game. They are amazing swimmers, lack an oxygen stat (they can stay underwater indefinitely) and have a tail-whip ability that can stun nearby enemies underwater, making them great for exploring the oceans without fear of a megalodon attack. Unfortunately, without any levels in your own oxygen stat, you won't be able to ride them for very long underwater before beginning to suffocate. They harvest a lot of fish meat and some biotoxin, which can help with taming other dinos.