Replace the antenna. Unscrew and detach the current antenna if it's bent or rusted. Purchase and replace with a fresh antenna from a local electronics shop for as little as $6. Broken antennas will inhibit radio-wave transfers and prevent the radio from working.
Attach an air hose to an air compressor. Plug in and turn on the air compressor. Blow air into the crevices of the volume and balance knobs. Move the hose around the entire perimeter of each knob and blow out all visible dirt and debris. Dirt and dust can agitate boom box controls and result in device malfunction.
Replace old parts with those from sister boom boxes. Save any old boom boxes to trade out cracked tape decks or handles, for example. Parting-out old parts from other boom boxes saves money on fresh repairs. Contact your product manufacturer, if necessary, to inquire about replacement parts.
Take the boom box to an electronics repair professional. Ask the repairperson to service any broken tape deck belts, or control system or motor issues. Test the boom box before taking the device for professional repairs to help identify specific problems. Press play on the tape deck, for example. If the device activates -- powers on -- but doesn't turn the tape, it may only require a belt replacement. Bring the boom box service manual to the repairperson so he has precise instructions on how to handle your precious "antique."