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Is there any algorithm known for braiding ribbon knots? More specifically I need to braid a generic ribbon knot presented as boundary of a ribbon surface= union of two 0-handles and one 1-handle. (Unfortunately MO did not allow me to add a picture).

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If I understand your question correctly, the answer is in the paper

Lee Rudolph, Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helvetici 58 (1983), 1–37.

In fact, in Section 3 Rudolph proves that every ribbon surface is isotopic to one arising from a braid (in a very concrete way, explained in detail in the paper), which is a lot stronger than what you ask for.

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