2002 Chevrolet Silverado wiring

What you said implies the brake lights are on when you're not applying the brakes. Do you mean the trailer's tail lights go off, and the brake lights do not turn on when the brakes are applied? If that is correct, on an older vehicle you would find a broken ground wire in the wiring to the trailer. On newer trucks like yours, most of them have separate turn signal bulbs and brake light bulbs. One bulb doesn't do both functions like they used to do, but they still do on trailers. You have to use an adapter harness. A lot of trucks have those adapters built in now. You can determine that by finding "trailer tow" relays in the fuse box under the hood.

Did this problem start right after the trailer wiring was installed or were the lights working properly before, and this is a new problem?