Persian as in the culture? If so it was huge, not only was it the court language and culture, it was also the language of the arts and sciences, thanks in part to a massive influx of Persian immigrants (brain drain?) that migrated there 1500s.

If you meant Persian, as in the Safavids, then that was also considerable, or even huge if you were to define some of the acts the Deccan Sultanates (the Shia ones, such as Golconda) as acts of subservience. Salma Ahmed Farooqui (Comprehensive History of Medieval India) writes that the Qutb Shahi dynasty of Golconda, themselves descendants of the Qara Qoyunlu, would read aloud the names of the Safavids Sahs in their call to prayer, which is an act of subservience, one you'd to for the Caliph.

Now others have covered the cultural impact; indeed the Bahmanid Sultanate derived its ancestry from the mythical Persian king Bahman.

However, the political impact cannot be understated. When the Bahmani split into the four Shia states of Berar, Ahmednagar, Bijapur and Golconda, and after the Mughals rolled in, it was important to maintain ties with Iran simply because the Mughals laid claim to all India. When they diplomatically engaged with anyone (other than the Persians or Ottomans and to a lesser extent the Chinese) the Mughals treated them as feudatories, with tribute and duty in order.

The four sultanates would play ball, but also maintained the supremacy of Persia as a way to keep their practical independence, a policy that would eventually fail after Aurangzeb invades the South.

Source: Writing the Mughal World, Sanjay Subrahmanyam and Muzaffar Alam