Obviously, it's not simply illegal to have a business that exchanges currency - legal currency exchanges do exist, after all. But you'd have to be careful if you wanted to open a business that does this.
According to 31 CFR 1010.100, you are considered a "dealer in foreign exchange", and thus a "money services business", and thus a "financial institution", if you are:
A person that accepts the currency, or other monetary instruments, funds, or other instruments denominated in the currency, of one or more countries in exchange for the currency, or other monetary instruments, funds, or other instruments denominated in the currency, of one or more other countries in an amount greater than $1,000 for any other person on any day in one or more transactions, whether or not for same-day delivery.
You are not considered a "money services business" if you do this "on an infrequent basis and not for gain or profit", but you say you want to "open a business" that does this, so this exemption wouldn't apply to you.
If you are considered a financial institution, then you must comply with all sorts of anti-money-laundering regulations, including, for example, verifying the identity of your customers, and filing reports with the US Treasury. You would definitely want to hire a lawyer to guide you through the requirements.