What you NEED to have is a professional site evaluation. You can try calling your County Extension Agent and/or Soil Conservation folks and see if they can help you out. You’ve already paid for their services in your taxes.
Or you can all a contractor or two come out and give you estimates on what you want will cost.
NO ONE here can give you more than the broadest of generalities. The estimate won’t cost you anything and it will give you a “baseline” number to work from.
What you need depends on what you want. Do you want to be able to ride immediately after a heavy rain? If so, yes. Ideally you need a base that is compact to concrete consistency, then 2~3" of footing above. If your native soil cannot be compacted to such hardness, you will need a create a man-made one, and that’s when we start to talk about $$$. For flatwork, you need a comacted base that is about 4". Ideally. For jumping, you want 6" and above. You want to grade your base so the water can run off on top of it instead of sitting and soaking into your base.
Pull out your excel spread sheet, call your local quarry to find out the cost with delivery. Don’t forget to include the material for both the base and the footing. Depending on where you are, the delivery may be as expensive as the material itself. Then call your construction guy to find out what it takes to do that. This guy will need to have heavy equipment to do the compact work and laser to level and grade.
You will find out very fast whether you need something.
I would recommend going to some of the barns in your surrounding area and see what they have put in for their arenas. It will give you a good idea as to what works best for your area, factoring in weather, and what you want to do on it. Depending on what state and area of the country you are in will also determine what is even available for footing. Remember one of the most important aspects to think about is drainage. Make sure wherever you plan to build your arena that you have made sure that it will drain properly. That is one of the key ingredients to keeping good footing in your arena. I wish you the best in your project. Take some photos when you are done with it so we can see how it turned out. It is great to learn what works best for different areas.