How tall is the balcony rail? If it's waist-high, it seems you could put some sort of "privacy screen" on it, as I've seen on many balconies in both the US and Europe. Or would that be against the regulations that your building has on that sort of thing?
I can't speak for others, but part of the reason I like to be nude is the liberating feeling I get rejecting the arbitrary societal norm that human nudity is shameful. We are born naked, shower naked, bath naked, love naked, conceive naked and for women, give birth to naked live, naked. Arguably clothes have benefits and we do have arbitrary laws from state to state that you need to consider, but I would suggest you simply buy inexpensive privacy screens and then enjoy your balcony. If you have take precautions to block the wandering eye from the street, then anyone else it interested in seeing you and crossing into your private domain which you might find annoying or flattering so long as they are not a pervert.
Then other thing you might try is find a safe local near you where other nudist go (beach, resort, lake, trail, etc.). Take a friend (male or female) and be nude with other nudists. If you are like so many of us, your inhibition and fear of being seen will disappear in seconds/minutes when you know you are amount non-judgmental nudist friends.
I hope you figure it out. I have been a nudes for 30+ years and was introduced to the lifestyle by two women I worked with who ask me to join them at the beach similar to my suggestion above. I was worried about shrinkage, erection, competition, etc. for about 20 seconds and then I realized everyone else was nude and enjoying themself with or without me. My presence really didn't matter because it was not about sex, it was about liberty to be you.
Perhaps this is obvious, but get down to waist level and see what you can see. If you can't see anything, no one can see you. Note where you can and can't see, and what you can see. At higher levels the deck floor itself blocks a lot of the view from below. For simple sunbathing simply putting a towel on the chair/lounge rail may suffice- wear the towel out and then drape it.