Great question. Since responsive web was first introduced several versions ago I have seen this come up. One recommendation would be to test/view the banner in both responsive and standard before implementing it. This will allow you to see if any adjustments are needed to the image and/or the colors. In my examples below, I would likely change the header text color:
Another recommendation would be to chose a banner that does not have a location specific image, since this can change depending on the size of the page which can potentially cut off parts of the image. One example of this can be see in the images above, you can see a Don't Walk sign in the full version but it is not shown in responsive version.
Hope this helps.