Hindu philosophies, especially Vedanta, say there is 'the Self' which is the reason for the existence of our human body. However, science hasn't proven the existence of that. Moreover, according to science it is gene only giving us a particular shape of a body. So, on what basis do Hindu philosophies say that Atman exist?
"On what basis do Hindu philosophies say Atman exist?"
As per Bhagavad Gita, the supreme reality viz. Atman is said to be neither "existent" nor "non-existent".
BG 13.13 - I shall speak of that which is knowable (should be known). Knowing which, one attains immortality. The supreme Brahman dwelling under Me, is beginning-less. "That" is said to be neither existent nor non-existent.
"according to science it is gene only giving us a particular shape of a body."
Whatever our ancient scripts describe is actually a spiritual science (VignAna) itself.
I presume that you are referring to the modern Physics & Biology in this context. We should understand that they are ever-evolving. Whatever we knew 1000 year back, was changed 500 years back, which again changed 100 years back & so on. This is natural, because it depends on observation and inference.
There are people, who say that all the humanity came into existence primarily due to reproduction among male-female. Which is furthered into gene theory etc.
They don't believe in anything else due to short- sightedness. Such ignorant people are also described in Gita as of [partially or fully] demonic nature.
BG 16.8 - Without [any] truth, without [any] foundation, "they" call the universe with no Supreme; Without any mutuality, [it's] born from the purpose of what else but lust.
Hinduism believes in authority of prasthana trayi, Gita, upanishads and Brahma sutra. The 3 scriptures talk about atman as the primary subject matter. It is on the basis of this primarily and then reasoning as auxilliary philosophies develop. As to the nature of the Atman whether there are 2 or just 1 or personal or impersonal there are differing opinions. But all philosophies accept that there is a soul that undergoes reincarnation and a higher self called Ishvara who manages the universe as a sport. From this basic understanding opinions diverge
According to Taittariya Upanishad:
II-vi-1: If anyone knows Brahman as non-existing, he himself becomes non-existent. If anyone knows that Brahman does exist, then they consider him as existing by virtue of that (knowledge). Of that preceding (blissful) one, this one is the embodied self. Hence hereafter follow these questions: After departing (from here) does any ignorant man go to the other world (or does he not)? Alternatively, does any man of knowledge, after departing (from here) reach the other world (or does he not)? He (the Self) wished, "Let me be many, let me be born. He undertook a deliberation. Having deliberated, he created all this that exists. That (Brahman), having created (that), entered into that very thing. And having entered there, It became the formed and the formless, the defined and the undefined, the sustaining and the non-sustaining, the sentient and the insentient, the true and the untrue. Truth became all this that there is. They call that Brahman Truth. Pertaining to this, there occurs this verse