How do I fix a drawer whose bottom is coming apart?

Can you post a picture? Drawer bottoms are set into a slot. Is the slot broken out? Is the drawer bottom warped? Is the drawer itself loose and coming apart? These problems are fixed in different ways.

Usually the drawer bottom can be realigned into the slot on the front of the drawer and glued. You can also add a few small nails to reinforce it.

A quick fix is to flip it over and slide it back in. Some of the drawers in modern furniture have really flimsy bottoms. You can reinforce them, crossing duct tape on the bottom, or adding something stiffer on the inside. A permanent fix would be as already suggested, replacing the wood (which may be more cardboard than wood).

I would replace the bottoms. Most drawer bottoms if they are sagging are not real wood, it's some sort of cheap MDF board or particle board.

Here's a good video tutorial: https://www.youtube.com/watch?v=eYls_OGRnrU

Glue a piece of wood across the center of the base for support.

I noticed that the bottom had simply popped out of the grooves made for it. What I did was lighten the amount of clothes so I did not have to stuff them down, put the bottom back in the grooves and then I will use duct tape tp keep them in place if necessary. It will help just not having to shove the drawer closed when it is really packed to tightly.

I use carpenter's glue to slide them back into the drawer slots, and then use 1/4" flat balsawood pieces glued to the underside to support the drawer glued in place. You can get the balsawood in the hardware store or a craft store. Make a knotch in the drawer sides to glue the wood into the sides and across the drawer bottom.

Then don't overload the drawer with heavy stuff to cause it to fail.