Is there any way to prevent or cure spider veins?

No and Yes: There is no good way to prevent spider veins, nor is there any way to cure them. Sclerotherapy which involves injecting them with a solution such as sotradecol or polidocanol is a good way to treat them, but needs to have maintenance in 1-2 years as they pop up in other sites.

Control but no cure: As dr. Isobe said, yes and no. Sclerotherapy is really the treatment of choice when it comes to spider veins in the legs. There are a number of chemicals that can be used, including surfactants (also called detergents), osmotic agents (saline), and chemical irritants (glycerin). Lasers are used too, but are less effective on leg veins. No treatment is permanent, so periodic follow up is needed.