Some of the drugs have to be accessible on board the spacecraft. Perhaps there are cold modules where an additional supply can be stored but the formulation would have to be modified in some cases. You might ask how supplies might be safely transported outside passenger modules in the space exploration SE site.
The article refers to "terrestrial shelf-life data" with compounds stored under conventional conditions on Earth (perhaps refrigerated but often room temperature), not at near absolute zero. The effect of radiation is not the main question, it is formulation stability (for instance phase separation) and retaining the activity of the API which by degrading loses potency and generates unwanted compounds.
Like most engineering problems many issues are considered during formulation including cost, route and ease of administration, dose, potency, and stability. Like a $100 hammer (or is it 1 million these days?), the preparations might be reformulated to behave better under longer periods and more extreme conditions. Degradation decreases at lower temperatures for kinetic reasons (if not thermodynamic) but storing stuff at very low T might alter phase stability. It might be reasonable to take along ingredients that must be combined upon arrival to Mars or as needed along the way:
However, for inherently unstable drugs, preventive measures can include altering the dosage form in a number of ways. For liquid dosage forms, the hydrolysis of drugs is dependent on the presence of water, and thus storage of dry powders followed by reconstitution in water before dispensing minimizes hydrolysis. Similarly, if a drug is known to be susceptible to hydrolysis at room temperature, its storage in a cool place is advised, and patients can be counseled on this practice. Additionally, pharmacists should provide the correct labeling on the packaging.
Note also that this is not a pleasure cruise. Humans intending to undertake interplanetary travel are probably less risk averse than the average person. At minimum it is important to remain functional on board the spacecraft. Although mission failure due to an unstable formulation would be undesirable, some increased extent of API degradation under the duration of the journey might be acceptable, provided degradation compounds have limited side effects. Bureaucratic hurdles would have to be overcome (somebody would have to sign off on allowing violation of FDA standards).