No, The Velvet Underground is not a goth band, because they play Experimental Rock, while goth music consists of Gothic Rock and its derived genres, like Deathrock, Darkwave, Ethereal Wave, and Coldwave.

However, they were an important influence on the development of goth music and are respected in the goth scene.

If you want to dive into the goth music scene, some of the bands that might appeal to you if you enjoy The Velvet Underground are Bauhaus, Cocteau Twins, and The Sisters of Mercy.