I'm sorry, but Ayase Eri's answer is completely wrong.
Misaka did not have powers before coming to Academy City. Espers who acquire their Abilities without the help of Academy City are called Gemstones, of which we only know of a small handful. (Himegami Aisa aka Deep Blood, Sogiita Gunha the 7th Level 5, and a couple others.)
Gemstones came first. Then Academy City developed a method of artificially recreating this phenomenon (the Power Curriculum Program), allowing them to mass-produce Espers.
The real reason Academy City is so interested in a Level 6 Esper is because someone who achieved that level would also achieve SYSTEM - the 'knowledge of the the will of the heavens'. Basically, they want a Level 6 so they can experiment with/on them, the same reason they want any other Espers.
As for Judgement, you actually get a skewed view of the organisation from Kuroko. Judgement officers actually have no authority outside the school they are assigned to - Kuroko keeps overstepping her bounds. Normally, any dangerous incidents are handled by Anti-Skill, the adult-run organisation.
Overall, the role of a Judgement officer is more like a part-time civilian patrol officer and are often used in community services, such as cleaning garbage from the streets (when cleaning bots are unavailable), helping locate lost objects, traffic duty, etc.
And of course there are weird and dangerous things happening all the time. Half the point of the city is to give superpowers to children - you better believe they have forces to keep them in check.