There’s no standard approach to situations like this. Every crisis will have its particular nuances and underlying issues that need to acknowledged and accounted for in any crisis communications strategy you implement. And the impact of the crisis will be hugely influenced by how amicable (or not) the parting of ways was.
The basic principle, however, is to be as honest and transparent as possible about why letting the cofounder go was the right move for the business. Your stakeholders – customers, clients, investors, partners etc – will need reassuring that the cofounder leaving is a positive move for the company, irrespective of whatever short term pain it inflicts.
The startup that loses a cofounder and retreats behind a wall of secrecy will quickly become the subject of the rumour mill, which can easily spiral out of control and cause serious reputational damage to the brand.
With this mind, it’s imperative that a clear and decisive crisis communications strategy is developed and activated as quickly as possible. You need to seize the opportunity to establish the narrative, and control the message.