They have both contributed to contemporary psychotherapy without knowing.
There two major concepts in contemporary psychotherapy: “here and now” and self-responsibility. Being responsible of each and every event, good or bed, is a notion that psychotherapists would like their patients to gain. If you are robbed, that is your responsibility. You had to know that that street was not safe. You had to know that in that time of day, this area is not secure. You had to know that carrying that much money / valuables by you is not a good practice. Irvin Yalom puts out that if Sartre was living today he would say that the Iraqi invasion and hunger in Africa are your responsibilities because you could do something to prevent them both.
Psychologists today are no different than Sartre in that sense. Many books on psychotherapy talk about how you are responsible of a misbehaving parent, spouse, friend, society. This is not to advice to save the world. The idea is that you have the choice to avoid such situations. You are responsible for putting yourself into trouble or not. You choose what you will live and experience. You also choose how you feel about this and that.
This was the Sartre part. On the other hand, contemporary psychology is deeply affected bu Buddhist perspective. Here and now approach of solving daily problems or working on complexes that emerged from the past is highly similar to Buddhist teaching.
On any day you feel blue, both a zen master and a psychotherapist would say that this life, you have to live through your stress and anxiety. Thus you will solve it. Beyond that, they would also say that you should forget thinking about yesterday and worrying about tomorrow. Today, this moment is the most important to you.
Therefore, it would be accurate to say that contemporary psychotherapy relies really much on Sartre and Buddha.