Do these new insights into prime numbers affect encryption security? Quanta Magazine reports:

Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers [...]. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them.
Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9.

Do these insights make breaking keys based on primes more mathematically plausible in a shorter amount of time?
No, because these new insights only affect the discovery and patterns regarding finding new prime numbers.  In order to break existing encryption algorithms that rely on primes such as RSA, you'd have to have a breakthrough in discovering how to factor integers into primes.

Primes are used in encryption keys as the basis of their generation:  two large primes are multiplied together to form a very large number.  The difficulty of factoring this large number to discover its prime roots is essentially the protection the cryptosystem offers.  Since you already have the primes (the private key, essentially), you need not factor the number and can immediately decrypt it.

Since the new discovery only relates to patterns prime numbers take which may assist a search for even larger prime numbers than already known, and not in factoring multiples of prime numbers, this discovery should be inconsequential for cryptography.