What can I use to keep hand-painted wood from getting stained/dirty?

Did you seal it with polyurethane? Then it is easy to wipe down when needed.

Is it going to go outside? Use a good poly spray that's made for the outside. (You could also use a brush on.) Do multiple thin layers and that should make a cleanable surface. If inside, there's quite a few polys you can use. If inside you could also use a nice wax and buff it up. Good luck.

Where is the wood, and how is it being used?

inside, used as an entertainment center.

For sure try polyurethane. Make sure its water based if you don't want it to yellow on lighter colors. If its a darker color it doesn't really matter. Good luck

Use a wood sealer. - Varnish or Wax or Polycrylic