Internet access from library computers has been a source of contention across the country. The issue is not just one of defining what is or is not appropriate use of public computers -- quite a complicated matter in its own right -- but also one of technical and practical details. A few of these need to be pointed out here.
Making a computer "kid safe" is a matter of restricting which of the billions of web pages and ever-changing content a computer can see. This is a massive undertaking, and the only way a library can even begin to implement such a plan is to subscribe to a filtering service. This requires an ongoing budget allocation. That is, it costs money.
While there are a variety of filtering services using different techniques, it is critical to understand that none of them are perfect. All of them have the effect of limiting access to legitimate websites. While this may be fine for young children, for adults this is double censorship: it restricts the free speech of web content providers, and restricts the equal access of library patrons. The practical result is that the library can NOT apply the filters to all computers. This gets complicated and expensive. There must be separate computers for children and adults, and library staff must monitor who is using which machines. Computers might even have to be in separate areas, lest a child see a computer screen over the shoulder of an adult patron.
Even with filters, the parents of children using the protected systems probably still have to sign a waiver holding the library blameless for what the child sees or does online. This is necessary because the provider of the filtering service might sooner or later make a mistake, and because a patron might figure out how to hack around the filters on a particular computer. The library won't want to be liable for technical problems beyond their control.