How Can Large Veins Under the Eyes Be Treated?

There is a delicate procedure where the veins can be directly excised. This seems to have the best effect for the veins themselves. Lasers do not work well for large veins. When the laser is turned up high enough to ablate larger veins, the overlying skin often gets burned.
Often, veins appear because of hollowness under the eyes and drooping of the cheek. In our practice, we often treat this with a LUSIC cheeklift, a limited incision cheeklift with grafts of the patient's tissues to the hollow regions. Often this reduces the appearance of hollowness and veins.
Hypertonic saline or sclerotic agents are not recommended for this area since it drains into the cavernous sinus of the brain, and an injection could potentially cause cavernous sinus thrombosis, a potentially fatal condition, or embolization to the opthalmic vein, causing blindness.

The skin under your eyes is the most delicate of anywhere in the body. Because it is very thin and prone to damage, it’s very common to develop noticeable veins in the area earlier than other place on the body, like the legs. As a board certified dermatologist and laser vein treatment specialist, I often use the CoolTouch Varia to remove unwanted facial spider veins for my patients. The treatment is completely non-invasive and provides excellent results. 

You need to treat these veins with a great deal of respect. Typically improvements for large visible veins require lower lid bepharoplasty. Injecting these veins can cause very serious complications.
Have a complete evaluation by a board certified plastic surgeon or oculoplastic surgeon.
I hope this helps.