What business degree is needed to open a clothing store?

If you want to open the store yourself, you don't need any kind of degree, though you do need the money to invest. Of course that does not mean a degree wouldn't help you know how to run the store and not lose all of your money.

If you want someone else to invest in your store, you need to convince them that you will run the store properly and not lose all of their money. A business degree, plus a really good business plan that shows you know what you are doing would help.

If you want to get a job managing a clothing store for someone else, different stores have different requirements, you have to ask them.