There are several factors that can contribute to broken capillaries on the face. Prolonged sun exposure is a common cause of broken capillaries, but they can also be caused by genetics and trauma, especially after surgery. Broken capillaries can also become more visible as we age and our skin becomes thinner. Rosacea, a treatable skin condition, is often associated with prominent broken capillaries as well.
There are many causes for broken facial capillaries including aging, sun exposure, genetics and certain medical conditions such as Lupus. This is just a short list. There could be many other causes. You should see a dermatologist or vein specialist for an evaluation and treatment plan.
There are many etiologies for the development and recurrence of capillaries after treatment. Although you can't change your 'genetics', you can change your lifestyle and avoid sun exposure and use very high SPF. You can also get treatments for rosacea as there are many and depends on the extent and stage of rosacea - from topicals to antibiotics to laser. Mild rosacea may only need topical antibiotic creams. Moderate rosacea may need treatment with oral antibiotics. Aging also can thin out the skin and also make capillaries more apparent on the skin as noted below.