Personally I haven’t used a kickboard in a training session for 17 years. And for most swimmers or triathletes, I’d encourage the same for them. The only time I’d get people using a swim float is if they really lacked confidence in the water, to help encourage them to relax and get comfortable. But I’d try and get them away from using a float as quickly as possible.
The main reason that I don’t use kickboards while swimming or coaching is that it promotes a poor body position. Wherever your hands sit on a float, your shoulders will be raised out of the water somewhat (which can put stress on the muscles and joints).
As a result of this lift, your hips are likely to be lower, and feet lower still. For swimmers whose kick is poor, this can exaggerate knee bending or kicking solely from the knee, and also promotes rigid, 90° inflexible ankles. I’ve watched some swimmers trying to kick along with a board and go backwards!
What I’d prefer to get swimmers doing is streamline kicking on their front or back, with arms along the surface as straight as is comfortable. The more flexible you are through the shoulders, the tighter to your head you can squeeze your arms. In this position, with your eyes down, you can draw your belly button toward your spine. And keeping your legs as long as possible, you can make sure you kick from the hip up and down with small fast kicks. When you need to take a breath, push your hands down and chin forward, take your breath and get your head back down into position as quickly as you possibly can.
For those less confident, you could always take a small breaststroke pull with your hands just to help you get up higher. From this you can do a side lying kick that’s easier to breathe on (by turning your head sideways as you would on full stroke), and making sure again that your legs kick both up and down, and in a small and fast fashion.
If you really must use a kickboard, I’d encourage you to use the smallest board that you can find so it’ll have minimal effect on your body position.
As far as training sessions are concerned, 5-10% of most swim sets should be focussed on the kick – not to make you kick more when you swim, but to make any kick that you do more efficient.
You could do 4 x 50m of streamlined kick with 15secs rest in your warm-up; or 50m kick 50m swim as a recovery interval between fast repeats; or 200m as 25m right side, 25m left side, 25m on your back, 25m on your front. The only limit to a kick set is your imagination!