Finding a correlation between drop in call durations or customer satisfaction because of YouTube might be really hard to calculate. Rather an approach that worked for us was the number of times customers chose self help over call/email support. 

Typically a video in isolation will not do the trick. You will need some accompanying content which can be accessed on a helpdesk/knowledge base provisioned for the company. A simple feedback form added to the page, which asks a customer whether the article was helpful, can tell a lot about these metrics. 

I found this infographic on the importance of self help customer service pretty relevant.

One of our customers runs a contact center operation in India. They already had a consolidated view of the customer's call and email records. Their business transformation team is piloting our product to capture additional customer data from social as well. The idea is to have the same team get an omnichannel view of the customer while interacting with him/her. Customer express their feelings about a product/service either before reaching the contact center or after, and in both cases, we help them capture that data and view it alongside their customer data. A great way to ensure your customers are not being neglected on social.