What is the importance( in Common Law) of the provision of
Responsibility distinct from Liability and enabling people to seek the
cessation of an illegal act even if they have not been damaged?
Who would go to court if they have not been damaged? Are they simply
busyjobs and vigilantes?
The terminology you are using isn't quite how it would ordinarily be discussed in a common law system.
People might go to court before they have been damages when the prospect of damage in the future is imminent, threatened, or likely but not certain.
For example, suppose someone has possession of your trade secrets as a former employee and is imminently about to commence work with another competing firm. You might seek an injunction to prevent that from happening.
Or, if someone has threatened to assault you, a court might issue a protection order, prohibiting that person from contacting you or getting close to you and ordering that the person's firearms be confiscated before an assault actually happens.
Or, someone has had a demolition permit issued to destroy a building that they allege is their building and you allege belongs to you. You might seek an injunction or declaratory relief to prevent that possibility.
You also might seek a declaratory judgment as to property ownership or boundary lines before building a fence or a building on what is at least arguably someone else's property, which could cause you to incur legal liability.
Or your might seek injunctive or declaratory relief if a law is passed that prohibits you from continuing your business (e.g. providing abortion services) and you believe that the law, if enforced, is unconstitutional, so you don't have to put money or liberty at risk to determine whether or not the law on the books is valid.
There are also circumstances in which a court can determined responsibility before damages are incurred even though they definitely will be incurred if there is responsibility.
For example, an insurance company might bring suit to determine if it is required to provide a legal defense to a putative insured who claims to have a valid policy from them. If the putative insured really is insured, the court might find that the insurance company is responsible for paying their legal defense costs as they are incurred, but it wouldn't have to do so if it was not responsible.
The issues could also be segregated in a class action lawsuit. A court might determine for a class as a whole, for example, that a company was responsible for all costs associated with a distinctive illness caused only by its product, but might reserve legal liability damages determinations for a case by case bifurcated trial process for each person in the class alleging to have had that disease and been damaged as a result. The class action might also hold open the ability of members of the class of people who used the product so that a judgment can be entered against the responsible party if the illness crops up later in a member of the class.