Bootstrap's Inverse navbar

Presumably you're using a modern browser and you can use its developer tools to find the exact line of CSS that you would need to modify. Here's some help on using Chrome's Dev Tools.

Once you've found the CSS that you'd need to change, I'd suggest however that you not actually change the CSS in your Bootstrap CSS, but rather create a separate CSS file (or SASS or SCSS or LESS file if you're using one of those) to contain all of your tweaks and customizations only for the particular selectors that you need to modify. The reason being so that you can cleanly upgrade your Bootstrap CSS to future versions without having to hunt down and re-implement all of your customizations.

The class that assigns the background-color is .navbar-inverse

@media only screen and (max-width: 767px) {
    .navbar-inverse {
       color: #000000;
    }
}