Balance trainers such as Bosu balls are a very good form of strengthening exercise/activity following an ankle sprain. If you just had a sprained ankle, this is NOT suitable for you. For a newly injured ankle, the focus of physiotherapy is to prevent aggravation and to promote stability. In some cases, we might get out hand therapist to build you a custom brace or splint to protect your ankle.
On the other hand, if you don't have an ankle injury, or any history of recent injuries, and you're just looking to build up your core balance using balance trainers, you can consider Bosu balls - they can provide quite a challenge for you to do exercises on them, with a range of exercises that will challenge your core. If you're not too sure, feel free to check out the Bosu workbook as a complement and companion to your Bosu ball. There's a simple video here to give you a simple preview to how you can train with the bosu ball.
To be safe, you should visit your family physician or walk-in to a physiotherapy clinic to be assessed, and we can prescribe a physiotherapy exercise program to build your your basic strength, stamina, and stability before you go on to higher and more challenging exercises such as the Bosu ball. Note: the bosu ball is quite tough, and you should get clearance from your doctor and physiotherapist first, just in case.
Other ways you can do to build your balance is by standing on one feet for > one minute without falling, and progressing to more minutes. Alternate between each leg. To make it more difficult, you can choose to tip toe, or you can get someone to throw you stuff for you to catch, such as bean bags (note: not at you, but around you). If you have a spare mattress, standing one leg on it will be a good balance trainer exercise as well.