DO NOT USE MOISTURE OF ANY KIND ON YOUR PHOTOS!!! Even the slightest of dampness will obliterate the original images. Water plays an important part in developing film and after the final images are set, water will leave you with nothing but a white piece of paper.
That said, I don’t believe much can be done to restore original photographs outside of professional servies. (I’m not trying to encourage you to spend money in that direction, I just really don’t know what else you could do that wouldn’t run the risk of damaging a photo).
To clean the originals, I would suggest using a dry cotton swab or cotton ball to rub off the worst of the dirt. A dry microfiber washcloth (you can buy them wherever you get house cleaning supplies) might also be worth trying. Also, I’ve read that canned air (the kind they sell for use on computer keyboards, electronics, etc) can be helpful in removing excess dust or dirt. For torn, worn or bent corners, I suggest trimming the edges with one of those paper cutters used for scrapbooking to ensure straight edges and perfect corners.
As for the musty smell: After removing the most of the residue, I suggest setting the photos out in a single layer somewhere they can air out for a day or two untouched (you might have to do it in batches to save room) – preferably somewhere with circulating fresh air. After that, I suggest storing them in a box with a few dryer sheets or some baking soda (make sure neither actually touch the photos: crack open the box of baking soda or fold the dryer sheets and tape them to the side of the box). This should help with the worst of the odor. When they smell better, you could put then put them into a photo album or you could just store them in the boxes and leave the air freshener of your choice with them.
For some photos, it might be better to just scan them or have digital copies made to use for display instead of the originals, which you can pack away for safekeeping.
