In addition to choosing a correctly sized paddle board, one also needs a correctly sized paddle.
How does one determine the correct size of paddle for stand up paddle boarding?
Paddle size is unsurprisingly related to rider height. While different people will vary by a little in their recommendation there are three general categories of usage that will determine the amount to add to your height: General flatwater, surfing, and racing. I've included advice from Tower and Aquabound (two respected paddle makers) for each category. You'll notice their advice is very similar.
Aquabound: Rider height +8-10 inches (20-25cm)
The general convention for determining the correct paddle size is to take the rider’s height and add 9-10 inches... Make sure you’re measuring the total length of the paddle from the tip of the paddle to the very outer edge of the grip/handle.
Aquabound: Rider height + 6-8 inches (15-20cm)
If you’re going to use your stand up paddle board primarily for surfing, you should go a little shorter than this 9-10 inch add-on to your height. Something closer to 6-7 inches taller than your height would be ideal.
Aquabound: Rider height +10-12 inches (25-30cm)
In this case, it’s best to go a little longer than the 9-10 inch add-on to your height. The best recommendation for racers is to get a paddle that’s about 12 inches taller than the rider.
A brief tangent on the other factors in picking a paddle:
I think it is worth noting that in addition to picking the length of the paddle one must also decide blade size, paddle material, and adjustable vs. fixed length shaft.
The advice I normally give is that you should stand with your hand up in a relatively relaxed manner (not reaching as high as you can), and the paddle should come up to your wrist on the raised arm.
Since most paddles you'll encounter are adjustable for length you can tweak it for comfort but it gives a good starting value, and nobody is going to be breaking out the tape measure when pulling adjustable paddles off the rack in a club or rental situation.