Velarían comes from the verb velar, which means to watch over, to guard, or to stay awake for. It's a beautiful and somewhat formal word that conveys a deep sense of care and protection.
In the song, the singer declares, "Estos ojos que estás viendo, mi amor, sólo velarían por ti" (These eyes you are seeing, my love, would only watch over you). This expresses a profound devotion, promising unwavering vigilance and care for his beloved.
Most love songs promise the world. This one promises a pair of hands. "Estas manos que estás viendo, mi amor / Trabajarían para ti" (These hands you are looking at, my love / They would work for you), and then the eyes, which "Sólo velarían por ti" (They would only watch over you). That is the offer of someone whose wealth is his labour and his attention, and he hands both over without asking for anything back.
The gratitude comes first, though. "Eres una mujercita encantadora" (You are a charming little woman) is followed straight away by "Debo de estar agradecido con la vida" (I should be grateful to life), and then by thanks to God for giving him so much "Para sentir una mujer como tú" (So I could feel a woman like you). By the end the hands and eyes turn conditional: "Estarían ciegos, si tú no los quisieras" (They would be blind if you did not want them). Los Ángeles Azules keep the accordion bright underneath all of it, so a line about going blind comes out sounding like a compliment.