Though
most western riders do not wear helmets, they provide a considerable
level of protection and are recommended for all riders.
Helmets worn while riding should have chin straps. Aside
from the fact the strap will keep the helmet on your head in
case of a fall, your animal may be startled if your helmet falls
off while you're in the saddle.
At the very least, a cowboy hat is recommended, particularly
for trips into the backcountry. Though it doesn't provide
protection in case of a fall, it will block the sun and rain.