The white part of the eye, known as the sclera, does have small blood vessels running through it which provide essential oxygen to these parts of the eye. Generally speaking, these vessels are not very noticeable. However, there are several conditions which can make these vessels appear enlarged, engorged, and more noticeable. Taking care of this problem cosmetically requires taking care of the underlying reasons for the enlargement of the vessels.
One common cause of enlarged, reddened vessels is mechanical irritation to the eye or eye dryness. This can occur due to sun, smoke, or dust exposure. It is often also common in contact lens wearers, if the lenses are old, overworn, or not properly cleaned. Lens rewetting drops (for contact wearers) or artificial tears may help reduce the irritation.
Another common cause of reddened eye vessels is allergic conjunctivitis, or eye allergies (seasonal allergies, mold, mites, and other things). These will usually respond to anti allergic medications, either pills or eye drops, as recommended by your doctor
I would start by talking with your primary care doctor
. In the meantime, avoid overusing "anti redness" drops as these can actually sometimes cause a rebound of eye redness if you use them for more than a few days.