Sanskrit is ancient. Historians date it to before 1500 BC. It was originally a language used for the Hindu religion, then expanded into royal Indian courts, and eventually found its way into everyday use. Today, Sanskrit is only used in the Hindu and Buddhist religions. In this respect, Sanskrit shares the same fate as Latin.
Did Sanskrit tattoo designs exist in ancient India? No one knows now. However, we know that henna temporary tattoos have been used by women in India and the Middle East for many hundreds of years, and we also know that India was quite technologically advanced in ancient times (India was producing and exporting high quality steel when Europe could barely produce poor quality iron). So it is quite possible that Indians already had tattoos in their distant history.
Did Sanskrit tattoo designs exist in ancient India? No one knows now. However, we know that henna temporary tattoos have been used by women in India and the Middle East for many hundreds of years, and we also know that India was quite technologically advanced in ancient times (India was producing and exporting high quality steel when Europe could barely produce poor quality iron). So it is quite possible that Indians already had tattoos in their distant history.