Would calculating the nth prime ruin encryption?
No.  The only plausibly relevant major cryptosystems involving that are RSA-based cryptosystems.  A typical RSA prime is chosen uniformly at random from 1024-bit primes, of which there are $\pi(2^{1024}) - \pi(2^{1023}) \approx 2^{1014} - 2^{1013} = 2^{1013}$ possibilities, by the prime-counting approximation $\pi(x) \approx x/\!\log x$.  Even if you could efficiently find the 1762496654486650458301017412708452577168455321425338564295164204627851827809673950656701400197184589192372578810839228796442397889449007625305174377608286256232618793314981228756675796669238641660676263338471020306341778397911556529236704830386308649499298417719554769934355960770722959180457300530690137th one, for example, that wouldn't help you to guess which prime I picked.  You don't even have a hope of guessing which of the $2^{256}$ ChaCha keys I'm using, and there are a lot fewer of those.