There were some answers for a similar question, none fully satisfying, check this link from: https://www.reddit.com/r/AskHistorians/comments/3w7ue7/why_is_basque_an_isolated_language_even_though/ with an answer from /u/mszegedy and several links in that post.

There is no simple answer. For one, because there are many, and one of them must be luck/chance. Basque, as only surviving pre-Indoeuropean language in Western Europe (Western Europe in its widest sense), faced different challenges in different time periods, and so every period had its own answer, or better -- answers. Let's simplify the issue and postulate just 3 such periods:

1. Basque (better said: Proto-basque) surviving Indoeuropeans.

2. Basque surviving Roman Empire

3. Basque surviving through Middle Ages to Modern era.

4. Surviving Indoeuropeans

We know Celts invaded Iberia in historical times, but they were not necessarily the only ones. There are signs of the possibility of pre-Celtic Indoeuropean languages in Iberia. Anyway, as most mentioned reason of Basque survival in those links mention Basques living in an isolated mountainous region, note that the mountainous northern Iberia (Asturia, Cantabria) was Celtic speaking area at the time of Roman invasion, and only the easternmost part of mountainous area remained pre-Celtic speaking (Basque lands). So, they survived it, unlike for example pre-Celtic languages of Britain and Ireland of which we know nothing (like for example the language of the builders of Stonehenge).

2. Surviving Roman Empire

This was really a hard one. There were several unrelated languages spoken in Iberia prior to Roman conquest. One is Celtic branch from Indoeuropean language family mostly in Northern and Central Iberia, but we know of at least 3 nonrelated non-Indoeuropean languages: the ancestor of Basque, Iberian in Eastern Iberia and Tartessian in Southern Iberia. Just one of them made it. Several different and unrelated non-Latin (and non-Italic) languages of Ancient Italia didn't make it. Several different languages east of Italia throughout Balkans north of the Greek speaking area didn't make it, with only one exception – Albanian. As Albanian Basque language was heavily influenced by Latin. There are numerous Latin words in Basque (and Albanian, for example mbret "king" from Latin imperator). Check this list: Basque bake "peace" (< Latin pace(m)), errege "king" (< rege(m)), lege "law" (< lege(m)). Note that those are Ancient Latin loanwords, not Medieval Spanish or later, compare with Spanish paz, rey, ley, where intervocalic -g- is lost. Interesting example is Basque bake, all Romance languages (with exception of some Sardinian dialects) palatalized c (and g) before e and i with different results in different Romance languages, Medieval Latin pronunciation did the same, but Basque in this case retained original Classical Latin pronunciation of ce, ci syllables, Basque language so provides one of the proofs that Classical Latin pronounced the same consonant in sequences CA and CE. Besides, Basque through loanwords preserved some Latin words lost in every of its daughter languages (ie Romance languages), like barka-tu "to forgive" from Latin parc-ere, this Latin word didn't survive in any Romance language. Basque made it through Roman times, it was heavily influenced by Latin, but it made it through.

3. Surviving later times.

Here were different challenges, the last one being Franco regime, but in a way, if you made it through Roman Empire, in comparison these challenges were piece of cake. This is more a question for an anthropologist, but it seems that Basques at least in this later times have a very strong self-identity and self-awareness and are not just linguistically but also culturally different from (there are several examples for this claim) their neighbours even in modern times. This mentality of "us vs them" is seen in the language, there are two roots in Basque that encompass ethnic diversity of our planet, one meaning "Basque": eusk(al)- and the other meaning "Non-Basque" erd(al)-, for example euskara "Basque language" and erdara "any other language", and Euskaldun "Basque person" and Erdaldun "Spaniard, American, Inuit... you name it."

I know I didn't answer your question, but I hope I at least showed why this question can't have a straightforward answer.