Cotton may work fine if they aren't carrying too much, but once you start putting towels, lotions and books in there, it might get too heavy for cotton.
I would go with a cotton home decor fabric. These are usually machine washable and soft, but also hold up well. If you use heavier fabric you also have the added bonus of not having to line or interface the bag.
Here is my easy tote tutorial that can get you started:
http://sewmamasew.com/blog2/?p=514.
You might want to make yours larger for the typical beach bag size. I've made myself some that are huge, but I love them!