Alibaba is notorious for allowing illegal goods from China to be shipped to the US by offering it's procurement service to US businesses. If Alibaba wants to control what it imports, it needs to put controls on its system, and those controls already exist. Also, Alibaba can actively work with customs and warn its customers that it is cooperating and willing to supply customer information to authorities when illegal activity is suspected.
From the moment the product is made to when it goes in the store (luxury name brand), there are too many "middle men" for Alibaba to keep the illegal goods off their website. Vendors on the Alibaba website will sometimes buy the products from distributor trucks and pay the drivers, but it is still illegal! It is my opinion there are just too many people involved for Alibaba to control it all. Too many are willing to do something illegal to make some money that are involved in the sell of illegal goods.
Alibaba is a gigantic success, however retail of illegal and inauthentic products continue without control. The problem cannot and will not disappear on its own without intervention by Alibaba management to monitor and control the products retailed on their website. The sellers are minting money from cheap illegal products and they won't stop unless some action is taken to prevent this.
No, the Chinese e-commerce giant Alibaba cannot get rid of illegal goods if they do not control what is available on their website, because they have no incentive not to have these things on their website. If there is money to be made, people will still post on the site until Alibaba decides to become self-policing.