In my time climbing, I have suffered through two ankle injuries and have heard that having a sprain puts you at risk for future sprains by weakening your ankle ligaments permanently to some degree. Ankle injuries are fairly common in climbing and sports in general. Therefore, it seems like prevention and ankle strengthening would be beneficial for any climber. I am putting some time into researching ways to strengthen ankles and will be testing them out. The goal is to eliminate future ankle injuries from happening.
According to the American College of Sports Medicine, properly applied ankle supports (tape, semi rigid, and rigid braces) and balance board exercising training can reduce the risk of re-injury by 50%. http://www.acsm.org/docs/current-comments/anklesprainstemp.pdf
If you have a wobble board or are willing to purchase one there are a number of activities that can increase your range of motion and overall ankle strength. These can be done sitting or standing and can progress to help all ability levels.
You can do isometric activities by pushing against an object such as a couch. If you are able to do these easily you can use a resistance band. For isometric activities with an object you can place your ankle down and in against an object for a 10 second count. Do this for 10 repetitions. Then do a up and out position against an object for 10 seconds, for 10 repetitions.
With a resistance band you can put the band around your foot and hold the ends with both hands. Gently push your ankle down as far as you can and then back into starting position. Repeat this 10 times. You can also tie the resistance band to a fixed object and then the ends around your forefoot. Start with your ankle pointed down and pull up as far as you can. Return to the starting position and repeat 10 times. With the band tied around the fixed object and the outer side of your ankle, you can also start with the ankle relaxed and then move down and in. Do this 10 times. Then keeping the band tied to a fixed object and the inside of your ankle, you can start with a relaxed foot and then move up and out. Do this 10 times.
Another simple exercise is to stand on a pillow with one foot and hold this position for a 10 second count. Repeat this 10 times.
All the isometric exercises recommended above are recommended by the American Orthopaedic Foot and Ankle Society.
You can also use stairs. Stand on the edge of the stairs and let the ball of your foot hang off the edge. Try to hold this position for 60 seconds, and make sure you are near a handrail so you can grab it if necessary. You can also stand on the edge of the stairs with your heels hanging off and hold this for 60 seconds if you can. Using handrails you can lower your heels, while on the edge, and hold this position for 10 seconds. Do 10 repetitions. Using the handrails you can also lift your heels and hold that for 10 seconds and 10 repetitions.
Slow and light barefooted jogging can help. Also swimming has been known to help.
If you are really solid with standing on one foot and balancing, you can try having someone throw a ball to you that you can catch while on one foot. If you do not have a partner you can bounce a ball against the wall.
Another idea is lateral jumps. You can jump over an object or line laterally. You could also try walking on uneven ground such as on stones.
These are simple exercises that anyone can do. You can make them progressively harder by adding weights or unstable surfaces. Strengthening ankles, particularly after having a past injury, can be very important.