First, what camera do you have?

Your camera can ONLY film 1080 X 720 right? That sounds like 16 X 10 ratio to me, so it's not too bad.
Can you film higher-res pictures by any chance? If you can, then simply do so, and then crop it down in post.

You're simply not going to be able to convert it very well at that size, because while you have enough pixels for the height, you're WAY short for the width. Therefore, if you did crop it down to 16X9 ratio, the video quality would be far short of HD. If it absolutely must be 16 X 9, you can add black bars on the sides. While the video itself wouldn't be 16 X 9, the editors and video players will think it is.

Unless you don't mind a drop in overall quality, you don't have many options. I recommend sticking with what you have (16 X 10-ish) since it's not that far off, and then upgrading your camera (If that's the problem) as soon as you can.

Oh, in that case, it'll be easy!
For videos I'll need to know your video editing software.

But for still pics, take them using the "Small" setting, then bring them into GIMP and click on "rectangle select" check the "Aspect ratio" box, put in "16:9" and then make the selection as large as you can without showing any white. (You'll be cutting off portions of the picture, but that doesn't matter.) Then go up to "Image" then "Crop to selection" and once it's done you'll have a quite large 16 X 9 picture. Then you go into "Image" then "Scale" and punch in 1280 in the top box, and 720 in the bottom box. (Alternately, you could put in "1920" and "1080" for 1080p HD.)