Normally the venue hosting the cover band will have a license that would cover that. This license would also cover music played from a jukebox, radio, etc. These would be BMI, ASCAP, and SESAC licenses.

The problem comes in where video is handled by a completely separate and different type of license.

You can record a cover and sell it with a "mechanical" license. This doesn't cover public performance or video as I recall.

Here are links that Christopher Schlagel had in another thread on the topic:
http://www.easysonglicensing.com/
https://www.harryfox.com/public/songfile.jsp

You are right...this is something you have to be aware of as a cover band.

I don't know how the venue could be held responsible for somebody that recorded video in the audience and posted it on the Internet. I would think the individual should be held liable...but the venue probably has deeper pockets. Wow!