In a anarchist world if there's no borders, how do we stop foreign plants, insects, animals and even diseases from spreading to places that could be harmful? For example certain plants and wildlife can be invasive, how do we stop this in an anarchist world?
Animals and plants usually stay at their natural habitat, if not relocated by humans. Legal borders usually have nothing to do with that as animals usually cross them freely (if they are in the one ecosystem) or artificially transported by humans for profit (which is not needed in anarchist society)
What about trade/export of vital resources? That is a huge source of introduction of invasive species to new areas and I don't see it dissapearing even in an anarchist society, since some vital resources necessary for things like medicine production simply can't be produced everywhere locally.
Can you give an example?
Like anything? It doesn't even need to be production related, supplying enough of medical equipment locally would be impossible on its own. Diagnostic equipment, surgical tools, gloves, syringes, various chemicals, drugs and medicaments... Like if you believe you can produce all of that locally, then idk what to tell you.
With modern transportation technologies it all can be transported without the effect of introducing invasive species. Rats, cats and pigs were introduced to new biomes because travels were insanely long.
How exactly is a modern technology going to stop rats from boarding the ship/plane?
They were infesting ships because they had to carry a lot of food for the crew, but now ships can hold much more cargo that is much stronger contained and they also have smaller crews and sail faster, so they don't need so much food. They don't even need to actually dock nowadays, everything can be transported by cranes in the container. Planes usually have too little free space for rats, so they can't infest it properly.