Allah tells us in the Qur’an that humans can use animals such as dogs for beneficial purposes by training them:
“They will ask thee as to what is lawful to them. Say: "Lawful to you are all the good things of life. And as for those hunting animals which you train by imparting to them something of the knowledge that God has imparted to yourselves – eat of what they seize for you, but mention God’s name over it, and remain conscious of God: verily, God is swift in reckoning.” (Qur’an: 5: 4).

The animals mentioned in the above verse include dogs as well every animal used for hunting – a hound, a falcon, a cheetah, etc.

We can infer from this that dog training is a service you can render and receive a payment for; for it is not different from any other skill that you acquire since it takes practice and effort. Therefore, you are allowed to receive a payment for your services in training and taking care of the dogs.