The "definition of insanity" solution
There appears to be a glitch in Hearthfire where giving your kids clothes can make them naked. The solution to this appears to be to give them more clothes.
It's possible that an unrelated glitch made them naked in the first place, and then giving them clothes made them naked-er. Giving them another set might be enough to fix the problem.
The "abandonment" solution
Many bugs in Skyrim can be rectified by leaving the area long enough that the area resets. For this to work, you've got to leave the house (and the general area - get far away for best results) and then take it easy for about 10 consecutive days. You won't win any parent-of-the-year awards, but you might get your kids clothes back on them.
The "put that down" solution
If you're playing on PC, you can try emptying out their inventory and then giving them clothes again to see if they'll wear them. As always, back up your save before playing with the console!
- Open the console by pressing the ~ key
- Click on the child.
- Enter the command
removeallitems player - this will put everything in the child's inventory into yours.
- Disable the child with
- Enable the child with
Now try giving them their clothes back, and seeing if they put them on.