Hi Paul, 

There are many chemical alternatives to moth balls, some of which do not have a smell. First off, moth balls do not reliably kill all life stages of clothes moths. Paradichlorobenzene (PDB) and dichlorvos pest strips are more effective, but they also have a strong odor, and they are synthetic insecticides. Deltamethrin and permethrin are more synthetic insecticides (derived from chrysanthemum flowers), but they are generally odorless and its efficacy against these moths has been verified. 

There are more natural products whose repellent activity has been verified. But I must stress that these are more repellents than insecticides; they may keep moths away, but they may not kill existing moths, depends on the dosage. This includes cedar shavings, camphor, citral oil and clove oil. Other essential oils have had their repellent/insecticidal activity verified by the literature, but not specifically for clothes moths (e.g. oil of lavender, eucalyptus, citronella, etc.). 

It is imperative to combine chemical treatments with other control methods. Starting with preventative measures, be aware of which items are made from animal fibers and are thus vulnerable (wool, suede, cashmere, leather, felt, etc.). 

These moths like dark, cramped spaces. Besides being in the closet, they may seek out dark, cramped areas to pupate. In heavy infestations, this can on objects near the closet (furniture, behind picture frames, etc.). If the infestation is heavy, inspect nearby areas that fit this description and destroy any pupae on site. 

Dry cleaning kills all life stages of the moths. After cleaning, clothes should be stored in sealed, thick plastic containers to prevent reinfestation, or at least until the infestation is completely eliminated. Freeze treatments can also be effective, if performed properly. Infested items must reach freezing temperatures for at least 72 hours. This is harder to achieve with older freezers that don't stay cold enough consistently enough. And bulky items will require longer for the internal temperature to reach freezing, requiring longer periods in the freezer.  

Lastly, there are pheromone baited traps for clothes moths. These traps are cheap and available at most department/grocery stores. I strongly recommend using those traps in conjunction with those other measures. These traps are a safe way to significantly reduce the number of moths infesting your items. 

Let me know if you have further questions.