Put simply, they become more verbose, Hawaiian is a good example, where the words can be quite long, even nouns E.g. - Kamehameha. When human languages lose complexity in one area they make up for it in another.
A strategy when a language loses case endings, possibly due to phoneme merger & loss is that this enforces stricter word order. As without the erstwhile case-endings the Subject(Topic) and the Object or Verb(Comment) cannot be easily differentiated.
Another strategy has been the use of tones for phonemic contrast the Pirahã language is a good example of this.
Note: You specifically mentioned minimal phonemic inventory, in case of a smaller consonantal inventory, one of the strategies used is to increase the number of vowels (E.g. - tense vs lax, length & diphtongs).