You have to be more specific as to what activity is legal abroad and illegal in the United States. There is no generic 'yes' or 'no' answer without knowing all the details.

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There is always the chance you could be prosecuted in the U.S., either rightly or wrongly, for the foreign activities of your foreign business. I don't know of any way you could ever be absolutely certain. You didn't identify the products and/or services you provide. I suppose one factor would be how morally repugnant your business activities are perceived to be by the authorities, which would affect the likelihood that you would draw heat. Insulating your ownership of the business through corporations and/or trusts might (or might not) be effective. Best to delegate day-to-day tasks and do as few things as possible to promote or direct the activities of this business while you are present in the U.S. Accurately report and pay U.S. income tax on your worldwide income.

You need to provide specific details concerning the "illegal" activity, then have a lawyer check to see if by statute the U.S. government extends jurisdiction over its citizens for offenses committed outside the U.S. It should be a fairly simple job to check.