Taking deep breaths is helpful.I also have used some mind deceiving self talk.Thing's like,I probaly won't get a shot,he isn't that big,he is just a doe with horns,basically any thing that takes my mind off the excitement that I may be feeling.
The best thing as others have said is experience,over time you certainly still feel the excitement but you are more apt to go on automatic pilot.
Another useful piece of advice at least it was for me years ago is to have your focus be on your form and/or your aiming point.Have your entire attention directed to putting that arrow or bullet in a very precise place,you know what happens when you hit that place.You get to spend some money at the taxidermist.
Another thing that just occurred to me is not to place pressure on yourself,hunting is supposed to be enjoyable,if you miss it is not the end of the world,learn from what you did wrong and get back at it.(I am not suggesting not being diligent or prepared)