The first priority is to protect your neighbor’s balcony while you find the source of the leak. Place a temporary waterproof tray or pond liner under the planter, lift the box onto pot feet, and check whether the dirty water is coming from drainage holes, cracks, seams, overflow, or a blocked or too-small saucer. Empty any tray before it fills, and never let runoff drip over the balcony edge. If water is escaping from the planter itself, a saucer alone will not solve it.
For a permanent solution, use a fitted drip tray with raised sides, sealed corners, and a slight slope toward an outlet. Attach a small drain spout and a hose that leads to a sealed container or a legal drain, not onto your neighbor’s balcony. Put a heavy-duty waterproof liner or pond liner under the planter and bring it up the sides to contain accidental spills. Seal hairline cracks with a waterproof planter-safe sealant or epoxy, and replace the box if the base or seams are badly degraded.
Reduce the amount of runoff by watering more carefully. Check the soil with a moisture meter before watering, use mulch to slow evaporation, and water slowly so the planter does not overflow. A self-watering or sub-irrigation insert with a closed reservoir can be a good option for balconies where runoff is a problem. Avoid relying on a gravel layer to “fix” drainage; it does not stop excess water. If the planter has drainage holes, water until only a little runoff appears, then discard the dirty water from the tray.
Also move the planter slightly inward from the edge, add a splash guard if needed, and check that the balcony can safely support the weight. Speak to your neighbor promptly, clean any stains, and explain the fix you are making. Check your lease or building rules before making drainage changes, and keep the system maintained so the problem does not return.