The bilingual Roman aristocracy noticed parallels between Latin and Greek. And once that aristocracy started conquering the Mediterranean world, Greek-speaking scholars took notice too, and began to suggest - with characteristic cultural self-assurance - that Latin was actually a dialect of Greek. The idea was famously (among classicists, that is) articulated by Dionysius of Halicarnassus, a Greek author writing in the time of Augustus:

"The language spoken by the Romans is neither utterly barbarous nor absolutely Greek, but a mixture, as it were, of both, the greater part of which is Aeolic; and the only disadvantage they have experienced from their intermingling with these various nations is that they do not pronounce all their sounds properly."^(1)

A number of other ancient authors - most, unsurprisingly, Greek - subscribed to what Benjamin Stevens (author of a very useful article on the topic) calls "Aeolism." In his Institutes of Oratory, for example, Quintilian notes in passing:

"[The study of Etymology] demands profound erudition, whether we are dealing with the large number of [Latin] words which are derived from the Greek, more especially those inflected according to the practice of the Aeolic dialect, the form of Greek which most nearly resembles Latin."^(2)

Examples of "Aeolism" could be multiplied further, but I'd rather point you to Stevens' article, cited below.

Underlying the theory of Aeolism was the idea that the languages of peoples who lived in close proximity tended to be similar. As our friend Dionysius says (when tackling the endlessly debated question of why the Etruscan language was so different from the languages of their neighbors):

"For, although it might reasonably happen, on the one hand, that men of the same nation who have settled at a distance from one another would, as the result of associating with their neighbors, no longer preserve the same fashion of speech, yet it is not at all reasonable that men sprung from the same race and living in the same country should not in the least agree with one another in their language."^(3)

Reasonable indeed, and a potential basis - if ever elaborated - for a broader theory of linguistic origin. Yet no such theory seems to have emerged.

Even the idea that Latin was descended from a dialect of Greek, though apparently well-known in some scholarly circles, was never universally current. Despite his acceptance of the idea that Latin was derived at least in part from Aeolic Greek, the Roman scholar Varro postulated different origins for parallels in vocabulary. Take, for instance, his discussion of the words used to describe livestock:

"Regarding cattle from which there is larger profit, there is the same use of names here as among the Greeks: sus, 'swine,' is the same as the Greek hus; bos 'cow,' is the same as bous; taurus 'bull,' is the same as tauros...This identity of the names in Latium and in Greece may be the result of invention after the natural utterances of the animals."^(4)

The words are the same in Greek in Latin, in other words, but that might just be because they're onomatopoeias (they aren't, incidentally).

To this point, I've focused on Latin and Greek to the exclusion of all else (that, after all, is what classicists do); but since you asked about other languages, I should emphasize that the Greeks and Romans had no conception of language families. There was a vague sense that some languages were older than others. Herodotus, for example, tells this curious story:

"Now before Psammetichus became king of Egypt, the Egyptians believed that they were the oldest people on earth. But ever since Psammetichus became king and wished to find out which people were the oldest, they have believed that the Phrygians were older than they, and they than everybody else. Psammetichus, when he was in no way able to learn by inquiry which people had first come into being, devised a plan by which he took two newborn children of the common people and gave them to a shepherd to bring up among his flocks. He gave instructions that no one was to speak a word in their hearing; they were to stay by themselves in a lonely hut, and in due time the shepherd was to bring goats and give the children their milk and do everything else necessary. Psammetichus did this, and gave these instructions, because he wanted to hear what speech would first come from the children, when they were past the age of indistinct babbling. And he had his wish; for one day, when the shepherd had done as he was told for two years, both children ran to him stretching out their hands and calling “Bekos!” as he opened the door and entered. When he first heard this, he kept quiet about it; but when, coming often and paying careful attention, he kept hearing this same word, he told his master at last and brought the children into the king's presence as required. Psammetichus then heard them himself, and asked to what language the word “Bekos” belonged; he found it to be a Phrygian word, signifying bread. Reasoning from this, the Egyptians acknowledged that the Phrygians were older than they."^(5)

Apologies for the unwieldy quote.

Neither Herodotus nor any other ancient author, however, claims that all other languages were derived from Phrygian. The general trend of philosophical thinking on the matter is reflected in Epicurus' Letter to Herodotus:

"But men, whose ideas and passion varied according to their respective nations, formed words of their own accord, uttering diverse sounds produced by each passion, or by each idea, following the differences of the situations and of the peoples. At a later period one established in each nation, in a uniform manner, particular terms intended to render the relations more easy, and language more concise."^(6)

Different peoples, in short, invented language independently.

There was certainly no widespread sense that Sanskrit was related to the classical languages, and nary a hint that the tongues of those uncouth northern barbarians had anything to do with the elegances of Latin and Greek.

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For further reading, check out this article on JSTOR:

Benjamin Stevens, "Aeolism: Latin as a Dialect of Greek." The Classical Journal 102 (2006/7), 115-144

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(1) Roman Antiquities, 1.90.1 (2) 1.6.31 (3) Roman Antiquities, 1.29.4 (4) The Latin Language, 5.96 (5) 2.2 (6) Diogenes Laertius, Lives of the Philosophers, 10.75-6; cf. Lucretius, On the Nature of Things 5.1028-40