What do I need to do to dissolve a business partnership?

I commend you on being aware that you need to do something of a legal nature to terminate your husband's business interest. If you do not terminate his business interest, you are right in that you may suffer consequences, as in being held liable to third parties, whether they be employees, other contractors, customers or the taxing authorities. What exactly needs to be done depends on the entity you are operating under and what the goals of both your husband and his brother are in terminating the interest. For that more facts need to be known, and I recommend that you seek legal counsel.