You can’t make anyone believe anything— not if they don’t want to—but you can change how you respond to your family’s criticisms. Most schools of Buddhism avoid proselytizing, and traditionally Buddhists are taught to not speak about the dharma unless someone asks. So consider making it part of your practice not to initiate conversation about Buddhism with your family.
If they bring up the topic, keep your comments simple, maybe just saying that Buddhism emphasizes awareness and compassion. You could point out that Buddhism focuses on ways to work with the mind, because that’s the cause of so much of our suffering. If they try to goad or debate you, practice mindful breathing and see if you can gently lead the conversation in another direction. In the end, the most convincing argument will be seeing the positive effect Buddhism has had on your life.