I want to add an individual sequentially numbered ID to each widget. In the template code, I have no problem adding the div with ID. However, I don't know how WP calls out a sequential variable. I can't seem to find any clear documentation on this.

For example, if you have three widgets in the sidebar, the source HTML will show:

<div id="widgetcontainer1">...content...</div>
<div id="widgetcontainer2">...content...</div>
<div id="widgetcontainer3">...content...</div>

I have no problem assigning a generic <div id="widgetcontainer"> in the template but I'm not sure how to add the automatic sequential numbering. I have some widgets that call out some messy class assignments, so this will help me simplify for a particular project.

Why not just use a plugin like Widget CSS Classes

That plugin looks like it would do the trick. I activated it, though, and I didn't see any additional classes added (first/last, even/odd, sequential).

Is there an easy way to do this manually?

if you read the doc, it says it adds a field to the widget that you can add other classes to.

I heard back from the plugin dev. I had to tweak my version of the theme to support it. Now it works great.