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

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.
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 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.