Does turning on the shower help reduce wildfire smoke in the air?

If the water drop collects the smoke particles while it travels though the air, then the water from your shower travels much smaller distance than the raindrops.

(Unless the fire is in your room, but then obviously you should aim the shower at the fire. And turn off the electric power before you do that.)

It's a different process.

Reference: https://news.mit.edu/2015/rain-drops-attract-aerosols-clean-air-0828

Indicates that a rain drop can attract "tens to hundreds" of particles (plus the one it nucleates around). Since rain clouds are about 5000 feet up, shower (or hose) water traveling 5-10 feet would be expected to grab 0-1 particle, depending on the ionization, which is also generated by the atmospheric generation method, which is not present in the "stream of municipal water separating into droplets"

The bracing freshness associated with post-thunderstorm air is often attributed to negative ions. Incidentally showers also produce negative ions, though typically as a side effect. I would think running the shower to clean air would be rather inefficient, particularly since ozone-feee negative ion generators exist as a consumer product.