I am aware that the minhag is often to abstain from going to the cinema, but does this have any halakhic basis. If so, does the prohibition extend to other video forms
According to this dinonline.org article, there is no prohibition at all to attend the movie theater during the Omer even if there is background music as part of the movie. There are many people who are stringent, though and don't attend movies during the Omer, and numerous that never attend a movie theater at all. (I'm one of them, but it's not b/c of stringency. I just hate oily popcorn odors :-)
The dinonline article has a link to a more extensive PDF that explains the source of the music abstention minhag and explains the various lenient and stringent opinions.
In brief, though, if you follow dinonline's leniency, then it negates your assumption and, hence its extension. If you can watch movies in the cinema, then you can watch it privately, assuming that the video has just background music and is not a music video - based on the analogy. However, you should read the PDF, as it has a separate discussion about whether you may listen to non-live music.