Yes—for almost any conventional urban planter box, drainage holes are not optional. Roots need air as much as water. Without drainage, excess water collects at the bottom, suffocates roots, encourages rot, fungi, and odor, and can turn the soil sour. A planter with no drainage may survive briefly, but it is hard to manage, especially outdoors where rain can fill it. The exception is a properly designed self-watering or sub-irrigated planter with an internal reservoir and a controlled overflow; even then, you need a way for excess water to escape safely.
Your second concern is just as important: drainage holes can leak dirty water. Holes do not guarantee a mess, but they do mean water will leave the box. If the planter sits over a neighbor's balcony, unmanaged runoff can drip through the floor, stain surfaces, annoy people, and create disputes. Soil particles, fertilizer, and organic matter make the water dirty. So the right question is not only "Do I need holes?" but "How do I manage the water that comes out?"
For a balcony, use a multi-layer system. Choose a planter box with drainage holes and place it in a sturdy drip tray or saucer with a raised rim. Better still, place the box on raised feet inside a waterproof tray, so air can circulate and the bottom does not sit in water. Make sure the tray is wider and deeper than the box and can hold the runoff from heavy watering or rain. Empty the tray regularly. If the planter is large, use a custom waterproof liner under the entire box and install a small drain or overflow pipe that leads to an approved drain or a sealed container—not over the balcony edge.
Water carefully. Use a well-draining potting mix, not garden soil, and water until only a little drains out. Watch the tray as you water. In rainy weather, check the overflow and empty trays before they spill. Add a layer of mulch or a geotextile filter over the drainage holes to keep soil from washing out. Keep holes clear, but do not drill extra holes without checking the planter's structure and waterproofing. For built-in boxes, line the inside with a waterproof membrane and create a drainage layer plus a controlled outlet. For renters or apartment dwellers, check building rules about water discharge and balcony modifications.
If you want extra safety, consider a self-watering planter with an overflow and a catch tray. These systems reduce runoff and keep water in a reservoir, but they still need checking after rain. Also avoid overfertilizing, because excess nutrients make runoff more damaging. A simple test: water the box with a small amount and see where the water goes. If you cannot control that path, redesign before planting.
In short, urban planter boxes need drainage for healthy plants, and they can leak dirty water if the drainage is not managed. With a drip tray, raised feet, a waterproof liner, controlled overflow, and careful watering, you can enjoy balcony gardening without sending dirty water onto your neighbor below.