Don't worry about padding anything.
I worked as the outdoor equipment manager for St. John's Cathedral Boys' School. We had 120 kids in the school with a 3 week canoe trip at the end of each school year.
The important thing about paddles: Don't walk on them. The leading causes of paddle breakage was either being stepped on, or putting a canoe down on them. The next cause was breaking when trying to pole free when stuck in a rapid or on a gravel bar.
Wooden paddle maintenance:
At the end of the season let it dry for 2-3 weeks. At the end of that time, give the tip a coat of linseed oil, making sure it soaks into the frayed fibers on the tip. If the shaft is rubbed free of varnish from contact with the gunwale, coat that too.
In spring give the whole paddle a coat of marine spar varnish on any parts that are scraped and dull. I prefer a satin or matt finish instead of a gloss as it gives better grip.
Most houses are too dry for long term paddle storage. This is true especially in northern climates in winter. Dry wood is more brittle.
I keep my paddles in the rafters of my open beam covered porch. A garage works well too.
At the school we kept the paddles in an unheated shed, stored in bins by length. Each bin was a box about 10" wide by 2 feet deep, and had a 2x4 frame about 4 feet off the ground. Paddles were stored tip down in the bit with the handles between the 2x4s above.
If you have a split in the blade, drill a 1/16" hole right at the top end of the split. This will reduce the split from continuing up the blade.
If you have a long split (5" or more) drill 1/16" holes about 1/2" away from the split 1/2 inch apart on either side. Using baling wire lace the paddle from the bottom much like a shoe, except that alternate crossings will be on opposite faces of the paddle. With each pass use a pair of pliers to pull the wire as tight as you can. If you have a lot of paddles to fix, having a jig so that you can clamp the paddle in a vice with the blade at right angles to the vice jaws will hold the paddle still while you pull, as well as holding the crack closed. I've seen wired paddles go through 3 more seasons.
A crack in the shaft just above the blade is terminal. If you have 6" or so of shaft between the blade and the crack start, you can wrap the shaft with braided nylon mason's line (about 1/16" nylon braided cord). Wet the cord overnight and wind it tightly. (Tie one end to a wall and you can lean your weight on the cord while winding the paddle) Tie off, let dry for 2-3 days and put 6 coats of spar varnish on it. This also makes for a nice grip. The winding should start several inches before and span to several inches after the crack.
If you are maintaining a fleet's worth of wooden paddles, having a hole drilled thorough the shaft just below the grip makes it easy to hang hang a 2x4 at just over head height with 2.5" finishing nails every 10" along the bottom edge,so that the nail shaft is horizontal. You can varnish the area by the hole, then hang that on a finishing nail Once you get them all hung, you can do them fast with a spray gun.