The problem is scaling. Donald Beaver, looked at the factoring numbers with DNA. $10^3$-bit number will require $10^{20000}$ test tubes by using Hamiltonian path idea of Adleman.
Adleman, also, observed that DES key search for 256 keys keys would occupy only a small set of test tubes. Let say $2^4$ tubes for $2^8$ keys. As a result we need $2^{56} / 2^{8} * 2^{4} = 2^{52}$ test tubes.
The previous statements are from previous DNA computing works directly on Cryptography. There are new studies on computing with DNA;
- There is a promising technique, called CRISPR9 with many version that enable editing DNA even at home.
- There is also interesting work called BLADE, where authors built 113 circuits from DNA.
- And, the DNS USB memory, MinION.
Under these and other new improvements on DNA computing, in very near future, one may come up with new results affecting Cryptography.