We’re planning on going scuba diving this summer under the supervision of a certified instructor as a cruise excursion. Unfortunately, my brother is worried he’ll run out of air (we’re only going about 15-20 feet deep, so on the off-chance that happens, it isn’t too hard to swim to the surface (plus, there are a bunch of people who’ll be there just in case; it’s the clear waters of the caribbean, so they can see him). Anything I can say to reassure him that he isn’t gonna drown?

From what it sounds like, you do not fully understand the danger of SCUBA diving. Even at 15-20 feet, with the air that you will be breathing, coming up too fast could do some serious damage. Do not dismiss the dangers because you are "shallow." Even though running out of air is unlikely, worse things can happen.

One of the most important things to having a safe dive is confidence. It is very easy to freak out down there and that is when bad things happen. If you brother does not feel up for diving, forcing him to do so is dangerous.

There are considerable safety measures in place. Knowing the specifics might ease him.

You are never alone. There is always someone there to help you.
Every diver has a second re breather so if you run out of air, you can grab your buddy’s spare and surface safely.
Before you dive, you take a considerable amount of time going through safety procedure and equipment.
Every diver has a pressure gauge telling him/her how much air he/she has left.

You might be able to get your brother to go through the training without committing to dive. Once he sees how it works, he may change his mind. But please, to not pressure him into something he is ultimately not wanting to do – especially with the danger so high.