Veins under eyes; what should I do?

The treatment of veins under the eyes is a bit controversial.  You have several options (other than doing nothing, which is always an option, as this is clearly aesthetic) - these options include laser, or sclerotherapy (which might be done with either liquid and foam), and microphlebectomies (taking the veins out thru tiny incisions).  Each treatment has relative risks, advantages and disadvantages.  With good eye protection and adequate skin cooling, an external YAG laser (1064nm) such as the Cooltouch Varia is an excellent choice - but it should be done by someone with a really good machine and plenty of experience.  Many physicians swear by sclerotherapy (injections of the veins) - this carries the very rare risk of blindness and  should be done with caution.  Very large veins can be treated with microphlebectomies. Whoever is doing your treatment should be an expert in whatever modality they use.  Hope that helps!

Under eye veins can vary in caliber and length, ranging from bluish varicose veins to light red spider veins. The treatment of larger varicose veins is very specialized and I have demonstrated my technique on the Doctors television show which can be found online. Treatment includes surgical removal with initiation of a skin brightening regimen to optimize skin thickness and collagen content. 
Associated dark pigmentation on the lower eyelid skin can be due to melasma, bluish veins, hemosiderin deposits, and hyperpigmentation.  
Topical regimens include: 
1.MelaClenz
2.Melarase AM
3.Melarase PM
4.Melapads
5.Replenish Retinoid Cream
Additional therapies for undereye rejuvenation include fractional laser, fractional RF, Covalent peels, and LED microneedling with Replenish serum. The video demonstrates an analysis of varicose veins.
Best, 
Dr. Karamanoukian
Realself100 Surgeon

Laser treatment is one option that has proven effective for veins around the eyes. I would suggest consulting with a vein specialist who has experience treating veins around the eyes before considering any other treatment options.