For republican Rome: The majority of romans who fell on their sword did so because they were on the losing side of the civil war, and some when they had lost a battle they should have easily won (although this rarely led generals to kill themselves) or because they were going to be successfully prosecuted and their career or life was shortly to be over. To answer the question, mostly nothing would happen. If it was a civil war, the sons of those who killed themselves would often be targeted for removal from public life, but that would often be all. The wives of these men would go back to their pater familias (head of their household) and would bring their children with them. I'd also like to note that this is a highly simplified answer due to the complex nature of when and why men would take their own lives. Sources: Tom Holland, Livy, Adrian Goldsworthy

For Imperial Rome: They would more often then not just be killed. The cutthroat world of imperial Rome meant that when a Usurper was defeated and forced to "fall on his sword" (although they'd normally just be killed outright by a member of his staff trying to gain favor with the apposing claimant) his entire family, friends, and political connections were dangerous to the life of the new emperor, and would be targeted for execution.
source: Adrian Goldworthy "Why Rome Fell"