You can follow these steps to get all 3 across the river safely:

Take the monkey across the river first, leaving the tiger and the banana on the original side.

Leave the monkey on the other side and go back for either the tiger or the banana.

If you pick up the banana, you can give it to the monkey on the other side and take the tiger across next.

If you pick up the tiger, you can bring it across the river and then go back for the banana.

Once you have the banana, you can bring it across the river and reunite it with the monkey.

In either case, you will have successfully transported all three without any of them being eaten.