what does that have to do with anything? Unemployment insurance is not due if, and only if, you agree with the circumstances of the unemployment. As an employer myself, I just don't understand the question.
You pay the insurance premiums. The State of Florida pays the benefits if the employee qualifies. If you believe the employee was fired for cause, you can appeal the ruling granting benefits. The only real advantage to appealing is to keep your premium rate from going up slightly. Unless you have lots of employees, challenging unemployment denial may not be worth your time and effort. You will have to show misconduct connected with work to convince the state to withhold benefits.
Ms. Ballman's answer is right on the money. I am not licensed in Florida, but at least in California the decision of whether an employee is entitled to unemployment benefits is made by a state agency, not the employer.
The real answer is, because you're required to. But I know that's not what you're looking for.
As the previous responders have said, it's not your obligation to pay the benefit or to make a determination of eligibility. BUT, as they also said, you can challenge a State determination if you feel it's worth the time/effort/money. What it comes down to, though, is that if the DUI was on the job, such as if the employee is a cab driver driving drunk, then he shouldn't collect. If it was on his own time, then it doesn't matter.