How do i safely get rid of this blood vessel under my eye?

Try eye creams: Although they are not very effective, some cosmetic eye creams can temporarily constrict the blood vessels in the eyelids. The only effective method i surgical removal, but this can cause more issues than it resolves. A concealer make-up is the safest method.

Laser: You have the option of having these veins treated with a laser such as the 1064nm long pulsed laser. I also know of some physicians that will inject with a sclerosant. I tend to use the laserfor these.

It depends: If the blood vessel is on the eyeball itself, laser or sclerosing treatments might lead to problems with circulation in the episcleral veins so i would avoid that. If it is on the eyelid skin under the eye, then longwave laser can be safely administered.

Laser or sclero: Depending on the size of the vessel and its location, many vessels under the eye can also be treated very safely and effectively with sclerotherapy.