How can I reduce the file size of a web banner image with Photoshop?

1. Save For Web
2. Set to JPG
3. Reduce image quality

If you're saving as a JPEG, you'll have the option of adjusting the export quality, which has a direct effect on the file size.

In adobe photoshop version CC14.2.1, CC2014 and CC2015; once you open the file in photoshop then got File>Generate>Image Assets to create a folder on your web files. then to resize it with saving  go to the layer just rename depends on file extension you used .jpg, .gif. or .png so by renaming the layers of banner and resizing too. just do like this Banner .png and the layer, Banner_4 as Banner_4.gif.

or for multiple size of banner 

Banner_4.jpg, Banner_4b.png, Banner_4c.png