Proving conspiracy is very hard to do because by its very nature any participants tend to keep their involvement secret. Besides conspiracy to board a vessel for the purposes of protest is hardly most wanted territory.
Actually conspiracy is often not all that hard to prove. Especially if a group has a media component to it.
If the actions are those of a lone individual or small group (3-5) the conceivably they might buy, lease of steal the boat required to get to the security vessel, with things paid for in cash or using cheques, cards and lines of credit belonging to the individual members.
However, if the group or organization has property or equipment (like boats) and the boats are used for an illegal purpose, then either the group is supporting the illegal actions, or the people who utilized them for illegal purposes did so without proper authority and the group would then need to proceed with legal actions against its own membership.
Given the abundance of audio and video recording devices, and the tendency for protect groups to record their actions for later broadcast to attract attention and support, it could be quite easy for an investigation to turn up people discussing plans to locate and track a vessel to be boarded. Then steal/lease/buy the required equipment to board the vessel, etc. Unless everything is done in person and in cash, there should certainly be a data trail which can be traced.
Given that there is a group or groups which are attempting to force Australian foreign policy, I could see how Gov't might be interested in taking action.
As for shutting down an organization... If the organization functions to aid illegal actions, then the organization itself can become an accessory to the illegal activity.
The easiest why I have of illustrating what I mean is this:
A used car parts company (legal company/organization) regularly receives stolen car parts. If the business owner/staff are aware of this occurring, then the business could potentially be shut down and/or the owner & staff could be in legal trouble as well.
In the case of the three men who boarded the Japanese vessel, if the environmental group arranged transportation for the men to the Japanese vessel, then the group could be held responsible for the illegal boarding by the men.
There are of course a number of different factors which can come into play (International law, Australian or Japanese law, etc)
-Cheers