Even a very young dog can learn to travel in a kayak with their owner. You should be careful, however, of an especially rambunctious puppy, as they may have a hard time sitting still and can end up jumping out of the kayak into the water, or tipping over you and your kayak, especially if you have a large breed puppy. If this is the case, you may want to wait until your dog gets a little older. Also, you will need to identify what issue you're working with: is the dog afraid of water, and needs to become comfortable not being afraid of the water, and the kayak? Or, does your dog love water and thinks that jumping out of the kayak into the water is a perfectly good idea? If this is the case, both you and your dog are liable to get very wet, and possibly injured if the kayak tips over. You will want to teach your dog to sit quietly in the kayak, usually in front of you between your legs, although some kayaks, such as sit-on or two-seaters provide an extra spot for a dog to sit. Or, if the kayak has a storage area on the stern or bow that your small dog can sit in, this may also be an appropriate perch for your pup. Most dogs are pretty excited to be going on an adventure with their owners and are pretty motivated to come with you and your kayak. Teaching them to be calm, unafraid, and that moving about is not OK on the kayak is the challenge.
It is a good idea to train in relatively shallow water close to shore, with a stable kayak. Recreational kayaks may be best for teaching your dog to ride in a kayak. Most recreational kayaks have a large space for their passenger that will accommodate both a person and a small to medium-sized dog. A large dog may require a two-seater kayak. Because the hard plastic of a kayak can be uncomfortable, slippery, and wet for your dog, you can put down a rubber mat where you want your dog to sit to give him better traction, or a folded towel to provide comfort. Doggy life jackets are also available commercially, and a good fitting life jacket for your dog may be advisable, especially if your dog is not a strong swimmer. If you are using a life jacket you will want to get your dog used to wearing it prior to training sessions, so as not to overwhelm your dog with new sensations and experiences. Take your dog for walks with the life jacket on for several days prior to initiating his first kayak ride. It is advisable for your dog to wear a harness, and have a leash available if you need to restrain your dog or handle him on shore.
Make sure your dog does his business before launching from shore, you don't want him stuck in the kayak looking for a place to relieve himself, and never tie your dog to a kayak. If the kayak tips you want your dog to be able to swim free of the kayak. It is also a good idea for your dog to have a good grasp of off-leash commands in case of capsize.