Insects carry infectious diseases, so I don't like them.
虫 insects
伝染病 infectious diseases/contagious diseases
運ぶ carry
ので so
好きではない I don't like them
「船で運ばれてきた虫が伝染病を日本にまき散らす可能性があるので嫌いだ」というのは There's a chance that insects that were carried on ships spread infectious diseases in Japan so I don't like them で表現できます。
参考になれば幸いです。