Use materials that will stand the test of time. Avoid cheap pressed board and thin veneers. Build your drawers out of solid wood, high quality veneered plywood or a similar strong manufactured panel.
Notch the inside of the sides and front of the dresser to accommodate the bottom of the drawer with a table saw; set the blade to a height of 3/8-inch. A drawer bottom that is inset into the sides and front will have far more support than one that is held in with screws or nails. For drawers that will be holding heavy objects, use a 1/2-inch or 3/4-inch piece of plywood for the bottom.
Use dovetail or finger joints for the corners of the drawer box. These can be cut by hand or made with a dovetail jig and a router. Drawers that are nailed together eventually start to come loose. Properly-fitted dovetails or finger joints that are glued together can last for centuries if properly cared for.