Languages are a part of history. Therefore, it is important to preserve dying languages for it tells of the past and preserves the culture of a people. The more we know of our past, the more we can learn from it and further build our future upon it. Languages are a gateway to the past.
Not only do linguists care about this, but the general public might, too. Understanding how language works can ultimate provide insight into rehabilitating aphasia, dealing with dyslexia, helping children with specific language impairment, helping older people with hearing loss and may help with degenerative diseases like Alzheimer's. If there are only a handful of languages to study, then we do not have enough data to really understand how language actually works in the brain. Language is an important way of establishing and maintaining cultural identity. This may not seem important to you, but to a lot of cultures and peoples, it makes them happy to not assimilate 100% to the dominant culture.
Ancient and dying languages are hugely important to the understanding of modern linguistics. They are the way we understand the building blocks of our own modern languages. It's important that we preserve them so we can continue to grow our understanding of them, as well as learning about how our own languages may still be growing and evolving.
Yes, we should do more to preserve dying languages. Even if these languages are no longer used by many people, they act as time capsules to past eras in human civilization. These languages can also provide insights into how the languages of the present have developed and spread throughout cultures.