From NHS Choices:
It's possible that it may help, however the common cold is generally a mild illness. Given that zinc carries the potential for side effects such as nausea and an unpleasant taste, zinc supplements are probably not suitable as a treatment for most people. Furthermore, large high-quality trials to assess the effectiveness and safety of zinc for the common cold would be needed before any recommendations could be made. It should be possible for a person to get all the daily zinc they need from a normal balanced diet.
WebMD seems to agree:
Zinc has become a popular treatment for the common cold. The evidence for zinc is controversial and contradictory. Some studies have found that zinc lozenges may reduce the duration of cold, perhaps by as much as 50%. However, many other studies have found no benefit from zinc for colds.
Zinc helps fight infection and heal wounds. However, if you already have enough zinc from your diet, it is not clear that getting even more -- from supplements -- has a benefit.
However Mayo Clinic says:
Recent analysis stopped short of recommending zinc. None of the studies analyzed had enough participants to meet a high standard of proof. Also, the studies used different zinc dosages and preparations (lozenges or syrup) for different lengths of time. As a result, it's not clear what the effective dose and treatment schedule would be.
Zinc — especially in lozenge form — also has side effects, including nausea or a bad taste in the mouth. Many people who used zinc nasal sprays suffered permanent loss of smell. For this reason, Mayo Clinic doctors caution against using such sprays.
My opinion after researching this:
In any case, it's probably better to just deal with it as it is generally not that bad, and without almost complete confirmation it may be a waste of money with unknown side effects. If it is very serious, talk to your doctor.