As an old dresser with wooden runners, however, there's nothing to keep the drawers from being pulled all the way out, and falling with a crash. Which could be a problem, since we have an inquisitive toddler. There are various approaches for making drawer stops, but to figure out what might be a good fit for ours I started by taking the drawer all the way out:

Many designs attach to the back of the drawer, but ours seemed kind of flimsy there and there was quite a bit of room on the sides, so I decided I would just put a screw on each side:

It works well! The drawers move freely but can't come all the way out anymore:

If at some point we need to remove the drawers to move the dresser I can take the screws back out.