This is really an issue, one of the reasons is improper perception of the term "invasive allien species". For many people it sounds like sci-fi jargon from E.T. films.
It is important to communicate first on a commonly seen example (some plant or animal which many people do know and have seen around) how it pushes away native species and how it may endanger functions of ecosystem, or (at least) autochtonous biodiversity - again, best on example of a species which will be pushed out, which people recognize, and ideally also like.
You can find quite a lot of information and also policy approaches to alien species on the website of the European Commission, its DG Environment.
I think that attention of the public media to this problem may increase when the invasive species suppose an important threat to the landscape, natural vegetation and even agriculture, and we have some examples of this in the Canary Islands, e.g. with the African grass Pennisetum setaceum. This is probably, at this moment, the best symbol of the war against the alien plants in our archipelago!