Treatment for Broken Blood Vessel Around Eyelashes?

I recommend that you see an oculoplastic surgeon who would give you the best guidance as to how to treat these. Excess energy can cause scarring of the eyelid margins and in the long run, cause undue drying of the eyes ... You can start with a dermatologist and follow up with an oculoplastic surgeon. 

Dear Freshblooms,
It is difficult to treat broken blood vessels on the eyelid margins because the laser used to treat these cannot be used in that area without risk to your vision (even if you wear eyeshields). Given your description, however, I must encourage you to visit a dermatologist and ask him/her to take a look at them as well as your lips, tongue and cuticles. You should make sure these broken blood vessels are not a sign of something internal going on (like liver disease or Hereditary Hemorrhagic Telangiectasias/Osler-Weber-Rendu).
Good luck,
Dr. Mariwalla