Left wood radical - the indexing component. Cups in ancient times were carved from wood, so the wood radical anchors the "drinking vessel" meaning. The same logic places 桌 (table) and 椅 (chair) in this radical family.
Right 不 supplies the sound - bù drifted to bēi with tone and vowel shift. Same phonetic in 杯 (bēi), 坯 (unfired pot), 胚 (embryo). The negation meaning has no role here; the rounded top of 不 may also have suggested a cup-rim shape.
