Putting Chapstick in your nose is perfectly safe and can help prevent your nose from drying out.
Just mark the chapstick with an N for nose so you don't use the same Chapstick on your lips to prevent spreading of germs.
You can also use Vaseline or Petroleum Jelly on the outside or inside of your nose to help keep the nose lining moist and prevent nosebleeds.
The Vaseline may also protect against sickness when used inside your nose.