This was recently profiled on the Doctors TV Show by Dr Raffy and his video from the show can be viewed on the link below.
Unwanted veins under the eyes are best treated with lasers. Laser treatments allow doctors to target specific amounts of energy to the veins in order to ensure that there is minimal risk to the surrounding skin. There are several laser treatments that are best suited for facial veins including Nd:Yag long pulsed dye lasers, Intense Pulsed Light (IPL) and Pulse Dye Lasers (PDL). Ask your doctor which laser treatment would be best suited for your individual needs.
There are several options to treat these veins. The very small veins can be treated by laser, micro needle radio frequency, VeinGogh, Vein Wave or even IPL. Obviously care needs to be taken for eye protection in this area. The larger blue veins can also possibly be treated by sclerotherapy depending on their size and location. You should see someone with experience treating veins in this area.
Eyelid veins can be improved with lasers.
The small vessels can be removed with intense pulsed light. Larger veins would need a 1064 nmYag laser. You will need to make sure your eyes are protected from the laser when doing this.
Some advocate the use of sclerosants. However, there veins communicate with the back of the eye so there is a theoretical risk of causing blindness from the sclerosant. Let me stress that this has never been reported as far as I am aware. Still the risk does not seem warranted. Yag laser is very effective on veins up to about 2 mm in size. After that, phlebectomy is a better option in my opinion.