The Bowery remains an open district of Arkham City throughout the whole game. Its northern part and the Museum area are more or less safe, as you will mostly come across weakly equipped inmates. You should however stay cautious while exploring the subway entrance area and the roofs of higher buildings, as usually they are patrolled by groups of armed enemies.
The only area of the Bowery you can't access at the beginning is the Arkham City Processing Center (screen above). It's found in the eastern part of the district and is becomes unlocked only after completing the Gain access to Wonder Tower main mission.
Batman Trophies - 39
Riddles - 11
TYGER cameras - 12 (4 challenges) + 1 camera Relay HUB
Catwoman Trophies - 5