There are a couple reasons why it's preferable to have the sides extend all the way to the bottom, rather than having them rest on the bottom. First, it looks nicer--you have one continuous piece all the way down the side. Second, it provides a little more structural support against racking. Whether you're using screws or nails and glue, attaching to the bottom across multiple axes (vertical for middle partitions, horizontal for the ends) adds strength.
That said, if you simply attach even a partial back to your cabinet, that will add all the racking resistance you need. Adding a face frame on the front will help, too.
You mentioned that you're using screws, and it sounds like you're mostly using butt joints for construction. If you don't want to be able to see the screws on the outside of the cabinet, you should consider using pocket screws and/or gluing and screwing solid wood cleats to all the plywood pieces. Screws run through the edge of a piece of plywood can split the plywood, but screws through the face hold better. Normal wood glue doesn't stick very well to melamine, so if the plywood is coated on both sides, you may need to use polyurethane glue (Gorilla Glue's namesake adhesive).