You could simply abstract the database code into a utility class, as you suggest, and this wouldn't be a terrible solution. (You probably can't get much better with WebForms.) Yet if the system is going to end up quite complicated, then you might want to pick a more formal architecture...
Probably the best option for ASP.NET is the ASP.NET MVC Framework, which fully replaces WebForms. It's built around the Model-View-Controller architecture pattern, which is designed specifically to clearly separate code for the user interface and backened logic, which seems to be exactly what you want. (Indeed, it makes the design of websites a lot more structured in many ways.)
Also, if you want to create a more structured model for your database system, consider using an ORM such as the ADO.NET Entity Framework. However, this may be overkill if your database isn't very complicated. Using the MVC pattern is probably going to make a lot more difference.