Sanskrit has been influenced in certain more fundamental ways than Dravidian languages have been by it
Well-known Indologist and linguist (Zvelebil 1975: pp50–51):
“Though the dominance of Sanskrit was exaggerated in some Brahmanic circles of Tamilnadu, and Tamil was given unduly underestimated by a few Sanskrit-oriented scholars, the Tamil and Sanskrit cultures were not generally in rivalry”.
However more recent research has shown that Sanskrit has been influenced in certain more fundamental ways than Dravidian languages have been by it: It is by way of phonology and even more significantly here via grammatical constructs. This has been the case from the earliest language available (ca. 1200 B.C.) of Sanskrit: the Ṛg Vedic speech.
Basically, Dravidian languages show extensive lexical (vocabulary) borrowing, but only a few traits of structural (either phonological or grammatical) borrowing, from the Indo-Aryan tongues. On the other hand, Indo-Aryan shows rather large-scale structural borrowing from Dravidian, but relatively few loanwords.