Implications of parity violation for molecular biology

The idea that there must be some reason that all terrestrial DNA has a right-handed twist (or D- vs. L-glucose, or whatever your favorite chiral biomolecule is), goes all the way back to the discovery of biomolecular chirality by Pasteur.

The connection to the weak nuclear interaction is apparently known as the "Vester–Ulbricht hypothesis," after its first appearance in the literature more or less instantly following the discovery of parity nonconservation in nuclear weak interactions. As Emilio's answer says, it's an interesting idea but it's hard to make the mathematics work out, just because the weak interaction is so feeble. The literature seems to go back and forth between "here's a way that the weak interaction could couple to biological chirality" and "there's no evidence for that coupling."

A review by Bonner (2000) concludes

Consideration of all lines of evidence leads to the conclusion that there is no substantiation for such a causal connection, and that the two levels of parity violation are entirely independent of each other.

But that hasn't closed the issue. Dreiling and Gay (2014) report the breakup of a particular chiral molecule by electrons has a small ($\sim 10^{-4}$) asymmetry in the electron polarization. Since fast electrons in the natural environoment mostly come either from beta decays or from cosmic rays (and the cosmic electrons are secondary or tertiary particles produced in the weak decay of muons), and beta-decay electrons tend to have "left-handed" polarization, this is the sort of asymmetry that could systematically suppress one enantiomer in a pre-biotic environment. So perhaps that puts the pendulum back to "maybe here's a way."

The reason that the weak nuclear force is called 'weak' is that it has a minimal role compared to electromagnetism and the strong nuclear force. Generally speaking it only really appears in nuclear decay, and while it can be responsible for the appearance of chiral ground states of nuclei (see e.g. Why are pear-shaped nuclei possible?) its influence on dynamics outside the nucleus is essentially negligible.

It's hard to fully rule out a weak-interaction origin for biological homochirality, simply because we just don't understand the latter, but generally speaking, very few people are seriously considering that possibility (at least, absent some as-yet-undiscovered linking mechanism).