Tanning booths and beds are private areas in which you can wear whatever you like. If you feel more comfortable in your bikini or swimsuit then by all means wear it. You can also choose to go more natural and wear only your birthday suit since no one will see you. Many women, however, do choose to wear underwear so that they do not burn the more private parts of their anatomy. You will also be required to wear eye protection since your eyes, even when closed, can allow the light into them and cause permanent damage.